Basically, the problem I was seeing is:

1. We move the LQT page (e.g. Talk:Foo) out of the way to "Talk:Foo/LQT Archive1".
2. We tell LQT to process the move (it does a limited amount of the processing immediately before we tell it, but we need it to do all immediately; see T95633: LQT conversion may fail for LQT talk pages with more than 500 threads).
3. We immediately continue and start copying data from LQT via its API (for this path we use a FauxRequest API call since it's the same wiki), but LQT doesn't see the content at Talk:Foo/LQT Archive 1.