If you have the circulation pump on all the time then having the bromine tabs in the basket should be OK. If the circulation pump cycles on and off then there is some risk. There's a reason bromine floaters exist.
If the top of the bromine floater is not sealed (i.e. is like a Trichlor feeder where tabs are put in at the top) then the gasses from bromine will oxidize the hot tub cover faster. This is why bromine feeders are loaded from the bottom and have solid tops that do not allow outgassing.
As for the discoloration, if the circulation pump was not always on then the tablets could still dissolve slowly and create very high bromine fairly low pH water near the floater and that can cause problems nearby. Even floaters in the main tub can risk degrading the acrylic of the tub if the floater gets stuck in one place. Again, this is more of an issue with Trichlor feeders in a pool, but there is still some effect from bromine in feeders as well. That's the nature of dosing concentrated chemicals that are strong oxidizers.
Normally specs of bromine shouldn't fall out of a feeder because normally the openings for the feeder are a little above the bottom so that small particles from bromine tabs should fall to the bottom of the feeder. As to whether any that fall out will cause problems to the acrylic floor of the spa, they might cause some minor problem because acrylic is reasonably strong, but it would be better not to have undissolved bromine tabs (or portions thereof) come in direct contact with the spa.
