It's really iffy. Mice and rats do best at temps around where we are comfy, so 65-70 is ideal. I don't have personal experience with this, but I've heard it several times that above 75 breeding is comprised (smaller, less frequent litters), at 80 breeding stops, and anything much higher is risking death.

Is there anywhere else possible to house them where the family won't be near often, basement or closet perhaps?

Also, I didn't noticed this at first in the pictures, but it looks as through you used 1/2 inch hardware cloth. If that's the size you used, you'll need to change it to 1/4 inch. Young mice can easily fit through 1/2 inch. You might to also add several more places where it attaches to the tub, as they can be stronger than they look and might squeeze through a spot that isn't attached well, even on the top. They can jump quite high!