Sounds like an upgrade to my cold towel idea
Slit the seat cushions and sink the tubes in. Very clean. I'd say exit the tubing under the seat to a "terminal block" of hoses. One block for "in" and one block for "out". You could consider using one of those 12V coolers instead of ice water or some fancy paltier thing. Put it in your back seat or trunk. I have grabbed a small 600gph bilge pump from the boating section. I'll let you know how well it moves water through small tubing.