Hi there. If you end up needing to remove a fixed cup on one or both bikes the Late Great Sheldon Brown's site shows how to make and use a very inexpensive tool to do that with. I made thetool and I've removed some really stubborn fixed cups and didn't have to use either heat or harsh chemicals to do. Here's the link to the tool which is show about two-thirds of the way down the page.
I've also have used large channel-lock pliers to both loosen and tighten a lock ring. Just be sure that you have something to hold the adjustable cup in position when tightening the lock ring.
https://www.sheldonbrown.com/tooltips/bbcups.html
Cheers