Swapping the mounted floppy disk image requires running commands from the DOSBOX command line, which I can't do while RECOVER is running.
So I wrote a python script to do it instead. I used WinImage to extract the CONTROL.### and BACKUP.### files into one directory, then my script reads all those files and recovers the archive.
I've used it on only two backup archives, so there may be latent bugs. Still, it worked well enough for my purposes, and put it up on github in case anyone else finds it useful. And at this point I fully expect someone to tell me I'm an idiot and there is a simple way to do what I want and there was no need to spend a couple evenings writing my own program.
Here it is: https://github.com/jtbattle/dos5_restore