You are re-caulking your bathtub. It has been a long process, but it is done. Caulking is a "must do" project to keep water from going behind your bathtub and wall tiles. The only question you have is: should you caulk around the bathtub faucet? Yes, you need to caulk around your bathtub's faucet. The caulking will keep any water and other liquids from going behind the faucet and into the wall behind the bathtub/shower assembly.
Caulking Around the Faucet
You need to caulk all areas around the bathtub faucet. No holes or cracks should be present around the faucet. If you leave an open area, water from your bathtub or shower may splash in the opening. Over time, the water will cause the area behind the tub to rot and mildew/mold can develop.
Breaking the Caulking Seal
If you need to remove the faucet later for repair or replacement, just break the caulk seal with a putty knife, screwdriver or other tool. Caulking is not permanent--it can be broken.
After applying the caulking, smooth it out with your finger or a caulk smoother (found at hardware and home improvement stores). Wipe off any excess caulking with a paper towel or clean cloth. To keep cracks from forming in your bathtub's caulking, fill the bathtub with water, stand in the tub and apply the caulking.
Try to use a continuous flow of caulking around the faucet. Do not stop and start. Use steady pressure and don't forget to let the caulk dry for 24 hours before using the faucet.