I don't have any real answers backed up by engineering studies or math or anything, but it's not unusual for a few drag race classes to run without a traditional cooling system at all. Especially examples like you mentioned that are short duration, and often if they are alcohol fueled. The race cars will be towed to the starting line though, so under your example it would have to be determined if that was allowed by the rules. But just running without a cooling system isn't treading any new ground, I'd guess any experienced engine builder who has done that combo could spec what was needed.
I would question the premise though, if just to save weight, for that short of a run the liquid cooling system needed is pretty small. A tiny radiator, a couple gallons of water, and a small electric pump is all that is needed, maybe a small fan. Small, light aluminum/plastic radiators are available at any pick-a-part salvage yard on the cheap if a budget is important. Your example of possibly coming up with some kind of air pump would most likely add back in the weight saved over a liquid cooling system.
Would the rules allow an air cooled engine? Why re-engineer the engine if someone has already done the work for you. I've seen people build some really light dragsters with snowmobile engines for example. Just throwing it out there, lots of possibilities, but what is or isn't allowed by the rules is the first hurtle.