Is it really that important to keep the vegetation warm? I would expect black nursery fabric (weed blocker) warming the soil to work better, but I'm still pretty new to tomatoes.

You keep the vegetation warm on a cool or cold night and you will see your plants accelerate past everyone else's and you will be picking sooner.

I've used slitted row cover before, My feelings are it does not allow for good ventilation on a hot day with little or no wind and it provides little or no frost protection.
You are better off with an ag fabric like covertan pro or re-may. They vent better and give you at least 2-3 degrees of frost protection.

As far as a heat sink goes, the wall of water surrounds the plant giving off some btu's where a 5 gal pail next to the plant is not going to have the same effect.