Maybe look into PipeWire, Reaper in jack mode works fine connected to PipeWire for me, without any of the problems you mentioned.

It is quite possible that your desktop audio already uses PipeWire depending on your distribution, you would need to look at latency settings if you want live playback.

I am not a lawyer this is not legal advice.

Search for copyright assignment agreements, there are a few good template documents. I’d request a signed document instead of simply stating it in the PR. In all cases I would recommend verifying the document and process by a lawyer before you start using it.

Also, I would consider not requiring copyright assignment if at all possible for the project in question.