seems, the cage gets exorted, too.

yes, that's an overdue function, and i'm surprised, it still is an issue.

if nothing else helps, you can hide the disruptive mesh parts temporarily in the parameters tab, but that makes the morphing kind of a guessing game and you'll have to work very carefully.

why spoil a nice, well elaborated new figure with impractical UV maps?

please do with the UV mapping as you have done with topology: simply make something reasonable.