Yes, that's true, I'm constantly making sure that ours is not getting in a mess.
Could you get a car seat heater and run it off a fairly beefy 12v power supply? Or is it more that the design won't fit your chair?
One thing I have is a jacket/body warmer that you can connect to a USB power bank. It actually gets pretty warm without power but honestly it can get very warm and a single charge of a decent power bank lasts a good 8 hours. You can get them pretty cheap on Amazon/eBay and they tend to be cheaper in the summer months for obvious reasons