Makes me wonder why it needs to feed on mice at all costs.

Did you ever try throwing in insects and leave the frog be?

I don't quit get the reason of tong feeding, a hungry frog will be able to get it's food.
If to much is left, feed a little bit less, if it's gone quickly, just throw in some more.

Seeing it does eat, but just doesn't eat what you wan't, i don't think there is much of a problem.
I don't like rodents as a staple diet as well.
You might try feeding it some more roaches, crickets, grasshoppers, waxworms, silkworms etc.
It's not bad varying it's food items.