We ended up utilizing multiple endpoints in a Premium FrontDoor, essentially acting like 2 front doors.

We made the first endpoint with a route of / and https://api.modernhosting.com/my-service/* as the only backend. This allows us to essentially "map" this new endpoint host (with no route prefix) to our desired backend with a route prefix.

We then made a second endpoint which has an Origin Group containing the first endpoint and https://my-service.legacyhosting.com/*, thus allowing us to direct traffic between the two.

I feel like this setup is a little gross, but it gets the job done. Since this is a temporary arrangement for us, I'm willing to live with it. Still, it would be nice to see Azure provide something that can handle this use case more gracefully.