Factors I would consider
1. Do I want to leave the PH with any - likely he'll only be interested in 375 so may over indulge in that calibre
2. Assume yr carrier will screw up zero on them all - difficult if the 500 is a double,
3. So for the 500 , 5 rounds apiece to ensure still all ok when you get there and build confidence and worst case for each buff 7 rounds each to down yr buff - 1 soft, hit but not down, 3 follow ups (solids) as he legs it, 2 more on a charge, longish shots pre charge/running away in the follow up, 1 insurance when he finally keels over. Call it 25 rounds all up - 5 softs, 20 solids
4. Leave the 338 at home, but know what yr 375 does exactly with Federal 300 grain TBBC and sledgehammers (should be same) at 50 - 200 yards and mount a variable scope , 1.5 - 6 (with Holland detachable mounts in case the 500 fails) and be able to shoot it over open sights. Assume Zero is really buggered when you get there and you need 10 rounds each, TBBC and Sledgehammmer to get it back. Assume the 500 also knackered when you get there and need the full 15 rounds sledgehammer solids for the 2 buffs.
Assume 25 rounds to down the rest and you'll leave the PH at least 20 rounds - suggest softs
5. Overall

.500 5 softs, 20 solids
.338 none
.375 25 solids, 55 softs