Can you recommend a brand of silicone rubber that would work for me?

Dow Corning, DAP, even hardware store brands like True Value. If it's
going to be damp you can get a mold and mildew formula. I steer clear of
GE because it caused me grief when I caulked in a shower surround and the
stuff didn't cure even after 4 days. Man what a mess that was to remove
and re-prep.

All these products would work, but I would include GE 100% pure silicone
adhesive and sealer. It is unpaintable, but tougher than all the paintable
types. Only trouble I've had with GE is when i use it beyond the "good
until" date printed on the bottom of the cartridge - then it indeed will not
set properly. Make sure whatever brand you use, that it is fresh (date will
be printed somewhere)
Roger