Hello.I've found that it is easier and quicker to just paint the whole thing and come back with some lacquer thinner and a paper towel and clean the little chrome thing.You have to be careful not to mess up the paint,but then,you should be accustomed to being careful if you have a pretty car. ::Good luck!
Vasoline and a Q-tip. Dab a little vasoline on a q-tip and carefully coat the ring. Paint the door, after the paint is completely dry, take a rag and wipe the vasoline off. I've done this many times on small areas. Easier and quicker than taping that small of an area.
You could also just paint the whole area as already suggested, and then use 0000 steel wool to wipe it clean. This will also polish the inner ring up nicely without scratching the chrome.
I painted a '66 grille using narrow striping tape on each chrome edge. Worked great. Don't think that is how they did it in the factory. I am about to paint the rechromed glove box door for the '66. Consider blue masking tape and a very sharp razor knife. Will let you know how it comes out. Good Luck.