Other ideas we've tossed around are refrigeration and food preservation, but the problem with those is that they need the power when they need the power, and so it's not exactly a way to sink excess supply.

It can still be a useful sink at small scales. You could make ice at those times of day if you're eventually going to need that ice later. It takes a lot more energy to chill something (especially water with its high specific heat and latent heat of fusion) that it takes to hold something at temperature in an insulated space. And then go on and use the ice later so that the need to chill something doesn't have to be synchronized with the exact moment in time you're drawing energy from the grid to run a refrigeration compressor.

Same with heating. Some smart water heaters can store thermal energy for later, too, and top off their energy usage for some times of day.

I'm not sure if the scale you're imagining makes these ideas too small to be worth pursuing.