Maybe a quick fix would be to stuff a beach ball in the cockpit before covering.
I currently have my Sunfish stored in the 2nd floor of my dad's barn with a fan on it - hoping to dry it out in the off-season. I blogged about it here.
But, for future outdoor storage, I had thought about using a piece of 1 1/2" or 2" rigid foam insulation, and cutting it to fit in the cockpit opening. That would solve the issue of the tarp drooping down into the cockpit, and prevent/deter possible critters from nesting in the cockpit over the winter, too.