How much do things move between two frames? Would you really have to recreate the full tree or just reorder a small number of leaves?
This is the first time of hearing BSP, and I read most of the OP's article to have a very basic understanding how it works.
Since this is a tree, reordering N elements would be approach N^2 complexity, would it not? (edit: I assumed you would have to find each node from the root, which could very well be a bad premise).
But this is one of those cases where asymptotic complexity seems a bit too reductive. One of the "n" in the complexity is really a fraction of the n.
That said, if I understand your use case correctly, you might not benefit much from having a binary subdivision of space for this. It sounds like what you need is more of a plain ordered list. One of the ways to implement that is as a binary tree, but an array that is kept ordered may be fast enough due to locality etc.
frame 0: open door
frame 10: still can't see past edge of door
frame 11: door opens enough to see big new area
Which gets into one of the major tradeoffs in rendering engines, do you try to minimize the average effort per frame or the worst case effort per frame.