I'm not sure I should comment because I am not very sympathetic to the ulterior goal, which aims at the Franco-English union essentially sitting on English colonizing capacity somehow and limiting the larger combined kingdom or kingdoms to playing just the role France did in the New World OTL, only also preempting any English competition whatsoever, or anyway whittling it down. And of course if England and France are fused together dynastically, French and English colonies can't go to war with each other without one of them being in rebellion against their shared monarch (or both are in rebellion). Thus perhaps we could have settlement very much as in OTL, only since by royal fiat New France and New England are two countries under one ruler, just as their homelands are, they must never fight each other. But you really don't want any English at all because even a stunted English settlement on OTL terms would disrupt and alienate Native peoples.
But first of all, England remains England. If your combination realm lasts from 1400 or whatever year it starts to the present day, I can see England getting more Frenchified than OTL, but there is no reason to think the English language will vanish, or that the culture will become indistinguishable from French. And being in a different setting--England being as you said poorer and smaller, it is going to be more hardscrabble. The English will want to go somewhere, a certain proportion would anyway.
What you want is for England as we knew it OTL to be removed from the colonizing pool somehow, or anyway diverted--if your focus is on Frenchifying North America, or more accurately having the general French pattern of light-touch contact via trade networks be the predominant one so that any intensely European settled and demographically dominated areas will be limited in extent I suppose, and the majority of the continent evolves from Native societies. If you don't care about other parts of the world then maybe the English can be diverted elsewhere, into the Caribbean or Central America or somewhere in South America maybe. Wherever the English don't go where they did OTL is an opportunity for some other European group with similar inclinations to the English, or bad enough if more restrained, so French power has to keep them out too. Best approach is to preempt settlement there by expansive groups by establishing settlement by nonexpansive groups--but being non-expansive, of course they will have little punch of their own, and the combination of their tripwire effect and overwhelming French power regionally must substitute for the tenacity on the ground that the OTL English settlements had.
I don't think a union of England and France is the way to get the result you want, or maybe you can if you settle for slightly different terms.
A monarchy that balances regard for English and French interests about equally, or in proportion to respective populations, will not want to resist English desire for settlement and expansion. The monarch must consider every category of subject or as you said, the unconsidered ones will rebel eventually. Until industrial transformations, England remains relatively poor, therefore her people are willing to emigrate.
Perhaps the kindgom as a whole can distract the English people with some alternative action that takes priority? Supposing for instance that the French adventure had diverted so much English attention that England's various rival neighbors in the islands--Scotland, but also a remnant of Wales, perhaps one that surges back from a rump state during a weak reign--oh, say, right around 1600 or so. Ireland--what if rouge Anglo-Norman lords have been becoming Irish-Norman and one of these families manages to unify Ireland in defiance of any fealty they may inherit to the English crown. So right in the period where OTL English settlement began surging into Virginia and New England, and spread form there, instead the English "wing" of the kingdom is bogged down trying to overcome opposition within the British isles--and any energies left over for colonizing are exercised in Wales and Ireland. The upshot is over a century or two, Ireland and Wales are incorporated and brought under the shared crown and very Anglicized, and Scotland might suffer the same fate in the lowlands, with the Highlands being claimed but hard to govern. This might divert English interests for a while and greatly delay projects for overseas colonization.
Meanwhile, assuming that the French branch of the kingdom included a limited, chartered company charged with developing profitable relations with the Native Americans, giving them an extra century or two to develop their web of alliances with no threat from English or any other settlers, then if the English belatedly, around 1750 or so, start seeking territories in America, perhaps given that Eurasian diseases are liable to decimate Native groups and that the French expansion pattern was not universally uniting--they would identify a rivalry and choose one side to aid, attacking the other Native group, it seems maybe there would be patches of territory on the Atlantic coast abandoned by decimated and defeated native peoples, and English settlers might be invited to settle well defined patches of land, while forbidden to expand beyond them. Since this prohibition would be part of the original settlement charter it could be accepted, at least by the first few generations, ungrudgingly. Meanwhile the pro-royalist Native allies will hit bottom and then demographically start to accumulate numbers, and absorb elements of European cultivation and other items of culture and probably intermarry with their English neighbors. Farther inland the same process, with much lighter intensity of contact but for a much longer time, has been going on and the interior gradually becomes a strongly Native cultured bunch of Metis.
I don't know if that is close enough to what you want. Note that in a rosy scenario for the monarchy, what starts out as a kingdom of two distinct peoples evolves into at least seven peoples--conquered Welsh, Scots, Irish and evolving Anglo-Colonial/Mixed and Franco-trader/Metis. This monarchy has a proclivity for adding national feathers to its cap.
I agree with RougeTraderEnthusiast too that a lasting Anglo-French union of any kind will have the interest of conquering or otherwise absorbing northward, up the North Sea coast from Flanders through Holland and on up to Denmark, if not beyond. At any rate the southern part of the Lowlands--reason being, as noted by RTE, England's border to worry about is the coast. Specifically the southeast coast is already covered as long as the union with France holds--to threaten from the south, some pretty long sea miles have to be traversed to avoid running afoul of French Atlantic based pickets. But Flanders is right there; in the hands of a hostile power it threatens both core kingdoms. Flanders must fall! And so on, albeit with lesser intensity, up the coast. Filling out borders toward the Rhine northeastward is the next priority, emphasized by getting control of the lower Rhine as well.
Is it ASB that Franco-England can do this? Well, first of all now France incorporates the strength of England, and is not distracted by English schemes against her. Second, OTL I believe France in the late medieval/Early Modern period did a lot of messing around in Italy, which from my naive view strikes me as optional. Conflicts with both Italian interests and Spanish seem likely to loom no matter how much France puts these regions on the back burner, and in fact it is natural for France to have Mediterranean interests and schemes.
I think I would lean on the English interest and English strength being added to account for the sewing up of the North Sea salient. I suppose acquiring Flanders will be the most vigorously opposed step, but if it does fall, then farther north will be held more weakly.
So perhaps the western part of the kingdom is tied down and distracted from transAtlantic adventures by the nearby adventures surrounding them in the North sea and the British isles.
If this scenario suits you, it is my best attempt at trajectory patching your proposed starting point and your intended ending point. If it does not suit the latter well enough--perhaps you need to consider that what you want is to nerf English settlement, and that perhaps incorporating England into France is too clever by half, and you might go for the simpler solution--the British isles must be crushed somehow.
Might a POD involving the defeat of Elizabeth by the Spanish serve you just as well or better? The Armada was hopeless, but what if the funds that went to the Armada OTL went to Parma's forces in the Lowlands, and he was able to surge forth and subdue all the rebels and incorporate Dutch shipbuilding, skilled crews, and most of all deny hostile Dutch remnants any base, so that the naval resources the rebel Dutch pinned Parma's forces to Europe with OTL are now available to try to escort them across the short strait to England instead. Escorting Parma's invasion was the Armada's function OTL--supposing Parma can get some long range guns comparable to the English guns, and put them on Dutch hulls in sufficient numbers to punch a hole through the defending English fleet, and Parma can land his armies, it is game over for Elizabeth. She might survive in exile but the conquest of the kingdom seems foregone. England under the hand of the Hapsburgs might be just the thing to prevent the English settlements.
They might also prevent France from setting up their trade empire in New France, but does it matter to you who does that as long as some European does? What if it is Flemings, or Spaniards, or even English adventurers, who do the French thing in the St. Laurence and Great Lakes region?