I have a specification sheet that says the load is 15 kW, 480 V, 3 phase. Is that 15 kW on each leg or 15 kW total?
I would expect rating in VA for anything that size.  Rating in watts implies the load is resistive e.g. heating (or otherwise has an inherent power factor of 1.0) and therefore no difference.  If that is incorrect, review your specs carefully. 

Figuring it straight: as Delta

That rating refers to total power.  For the clamp-meter current on any wire, you need to take 


  15000 / 480 / sqrt(3) 


Giving 18.04 amps. 

Under NEC rules, most loads get treated as continuous loads.  That requires a 125% derate, to 22.55 amps.  That will require 10 AWG copper wire based on NEC 240.4(D).  Generally you would expect a manufacturer to size the load at 13,000W instead, to snake under 20A continuous, and allow use of 12 AWG wire. 

Figuring it as "wye"

Your idea of 15,000 watts per phase is wrong.  However, you could think of it as 5000 watts per phase, but phase-neutral voltage is not 480V, it is 480V/sqrt(3), or 277V.   So now the math is different: 


  5000 / 277


Well, snap, we're right back at 18.04 amps!