I'm a little lost on what you're saying about the motor plate saying 15A vs the specs saying to run it on a minimum 15A circuit. The two don't jive. If the motor plates says 15A then it needs to be on a 20A circuit. 15A X 1.25 = 18.75A so 20A circuit.
Edit: I'd use the motor plate amps. The HP rating is most likely baloney.
The NEC does not permit using nameplate data for supply conductors.
You use 430.248
As eddie1278 pointed out to be NEC compliant you need to size it as 3HP @ 17A. Since the motor data plate has 3hp on it that's what you should go by. If the data plate only listed the amps then the amps is what you'd use. I was thinking the 3hp was just the advertised hp printed on the compressor.

You got it mostly right. Read 430.6(A)1 starting about halfway down. If a motor is marked only in amperes, you are basically supposed to make a "best guess" on the motor horsepower and then select the ampacity from the motor tables.
Ok for the armchair ec's
start at
334.80 then go to 310.16
I never stated a code section just the ampacity table 310.16. Type NM #12 in the 60 degree column has an ampacity of 25 amps. End of discussion.
By the way I'm on the 2008 code in PA