Update: I love you.
It took a couple tries to get my desktop and laptop connected, and I don't know why, but it definitely works.
I'm going to really miss clipboard sharing, but I can make do for now.
I don't think I mentioned it, but my work laptop is Windows 11, so I'm happy to report that this is working great even on Windows.
Barring that, something like a Google Voice number would work, too.