Mike; almost any kind of paint will do. Are you going to brush or spray? Whatever you paint on to chromium will almost certainly need an acid etching prime first. I suspect the word ACID may petrify you but with some careful masking up first, just cover the area you plan to paint. Do you have a picture of the bumper?
I have developed an in-depth illustrated description on how to paint the recessed lettering on a bumper. Go to www.fzoldcars.com and click on "Technical Stuff" at right hand top of Home Page to see detailed information. Sorry the bumper is for a 1939 LaSalle, but the process is the same and it works.
Several of the pictures are attached, but the specifics are in the description. Check it out - I have used such a method for many years, starting with restoring antique slot machines.
Fritz