I have had very good results using tape that is designed specifically for lasers.
I’m in Canada so can’t really order proofgrade materials, but have found that Trotec is a reliable supplier of laser-grade products. They have rolls of masking tape that are nice and wide, and easy to apply. (Make sure to adjust your thickness settings to accommodate the thickness of the tape.)
Personally, I would avoid regular painter’s tape.
Congrats on your new GF! Just a heads up, I’ve read that the original Apple bands don’t engrave very well, but the aftermarket silicon Apple bands do. So if yours is an original band and it doesn’t come out great, at least you’ll know why.