Considering OTL's Reconstruction and the violence that happened during it, I doubt the Radical Republicans would be that reluctant to force a new Constitution down the South's throat. They viewed the former Confederates as a bunch of rebellious traitors who deserved whatever punishment they got. Also, considering that the founding fathers made the process easier to ratify the Constitution than it had been to amend the Articles of Confederation (which required the approval of all thirteen states), a similar thing could be done with the process to ratify a new constitution since that precedent had already been set. As far as the risks, if the Radical Republicans truly came to believe the Constitution had failed, I see no reason to believe that they wouldn't think it worth the risk to create a better one or at the very least they would have sought more than the 3 amendments that were passed in OTL.
I hear what you're saying, and I agree that the Radical Republicans certainly believed that the Southern states were entitled to nothing. However, we're only talking about one wing of one party. Most Americans did not feel this way (not even Lincoln). In fact, the Radicals were regularly blocked from imposing the kind of harsh penalties against the South that they would have liked.
I am not even sure that even the Thirteenth Amendment would have passed Congress after a Confederate surrender let alone getting the necessary number of states to call for a constitutional convention.
I think this scenario isn't as far-fetched as people are making out. If the thirteenth amendment gets blocked in Congress, which could have happened, then a convention could well be on the cards. And after being completely devastated by the war, I can't see the Southern states rising up again on a procedural issue. That would only happen depending on what specific changes are made at the convention.
This is probably the only scenario where I could see a convention being called.
But let's consider the sequence of events that would have to occur for a Second Republic:
1) 2/3rds of the states would have to call for a convention. Remember... Congress cannot call the convention. It has to come from the states. So even once the Republicans gained the two-thirds majorities in both Houses of Congress in 1866, they were still powerless to call a convention (though they could and did pass new amendments). Many state legislatures would be resistant to the idea of a convention and especially if the many of the people in favor of it were Radical Republicans.
2) Second, the delegates would have to be chosen and the convention would have to be convened. This becomes quite a thorny issue because the Constitution does not specify who can be considered as delegates or how many delegates each state would be allowed. Some states may even boycott the entire process if the Republicans are handpicking southern delegates. While it seems trivial, this selection process could kill the whole project right there.
3) The convention has to actually draft a workable document. It is quite possible for the convention to leave completely empty-handed if they are hopelessly deadlocked. Because a failure to reach an agreement simply results in the status quo being maintained, conservatives have little inclination to reach an agreement. Contrast that with the 1787 convention where virtually everyone present agreed that the status quo was unacceptable even if they strongly disagreed over what should replace it.
4) 3/4ths of the state legislatures must accept the new Constitution. This next step depends largely on what exactly is in the new draft of the constitution. If the Radicals dominate the convention, you can be sure it will be roundly rejected by the South and the border states. More likely than not, a newly-drafted constitution in the wake of the war will include many caveats to the first, second, fourth, and sixth amendments to make dealing with future rebellions and overseeing Reconstruction more expedient.
If Lincoln is still assassinated in this timeline, it is quite possible the succession to the office of President would have been codified a hundred years earlier.
The potential for renewed civil war is not just with the crushed Southern states. Depending on the content of the new constitution, other states might be completely unwilling to accept it as well. I don't think this would be an issue though because obviously I think a new constitution would have been defeated much earlier in the process.