Dorman shows only steel plugs in slightly over and under size.
I've always checked their catalog, picked up some locally (usually have to order) and pick the one I like best.
here is their data for 360

362660.1 - 1-41/64 In., Height 0.360
555-033 - 1-45/64 In. SC, Height 0.270
555-068 - 1-41/64 In., Height 0.360
555-068.1 - 1-41/64 In., Height 0.360
555-072 - 1-21/64 In. SC, Height 0.300

sc is shallow cup that you may have, try the .36?