A carp can withstand some pretty long periods out of water and needs very little oxygen in the water, Two parts per million if I'm not mistaken.
I've seen ppl release carp into holding ponds they built to keep them alive for weeks in clean water prior to killing and filleting them,After an hour laying on the bed of their pickup.
Doubt keeping their head out of water for any short periods is going to have much of an effect on their fighting abilities..Love the things to much to ever think of killing em...!
Wish ya great fishing,Bill