I recommend just getting one of the no-frills affordable unpowered $40 bidets on Amazon. The idea of the water not being warm sounds much worse than it actually is, and after just a couple uses you hardly even realize the water isn't heated.
Personally, I think an upgrade to something in $100 range is worthwhile. I've got an original Tushy (cold-water) model that is easier to control the flow. The cheaper Amazon models tend to be either off or a sharply aimed firehose at least with my water pressure.
I have an unpowered one with cold water and I agree that it's not an issue at all. It's even refreshing!