The Republicans would not have called for a convention in order to redress its weakness in preventing a civil war. There are two reasons a constitutional amendment banning secession was never passed after 1861.
1. The Supreme Court eventually ruled it unconstitutional in Texas v. White, and again in Williams v. Bruffy
2. Passing such an amendment would acknowledge that the south had the legal right to secede beforehand. It's the same reason LGBTQ+ activists preferred to use the courts instead of the ballot box. If the court rules something, than it kind of retroactively legalizes the institution so the activists can say "we had the right along," rather than saying "we were given the right."
If what you are looking for in this second constitutional convention is something like a change in the system of government, say from a Presidential system to a Parliamentary System, you can achieve this with just a couple amendments.