Re-heating beurre blanc without splitting
I'm quite sure being able to reheat beurre blanc is a superpower without a superhero. Your efforts are commendable but I think inevitably futile, I'm sorry to say.
The problem is not that it splits while heating up. The problem is that the sauce splits while cooling down. This is caused by the butter solidifying. When heating it back up again, it will simply become visible that the sauce has split.
It's unfortunate, but you're probably best off not making too much beurre blanc. If you still have some left over, it's a great component to use in a cream-based sauce. Also, if you really want to make a beurre blanc that you can cool down, you could experiment with adding cream. But even though it might turn out to be a nice sauce, it's not the real thing.