Thanks for the advice.
To be specific, my new BB is a Shimano BB-ES25 and it is already slathered in grease but I may add my own White Lightning Crystal Grease which is waterproof.
I think I'll use the caulk method but first to seal the seat post but first, I am going to have to try to find some black colored caulk if such a thing exists since my frame is all black.