I might be wrong since I've studied this a long time ago, but from what I remember, in order to do that classically, you need to copy the output bits somewhere else before uncomputing your system and recovering the ancilia.

That's technically fine, as long as you have an infinite supply of stably initialized bits onto which to copy your result. Initializing those bits is going to be non-reversible in some way.

I haven't seen any reversible low power superconducting gate that can credibly operate at a high temperature - not because of the superconductor itself, but because of thermal noise. Again, I haven't read through the literature in this field for a while (and it wasn't that extensive either), but from what I recall what you're proposing is roughly as difficult as making a gate for a quantum computer, and you have to keep your system way colder than your critical temperature from that due to thermal noise. If you have any links for high temperature physically reversible logic gates I'm all ears.