Hey, singer and electrical engineer here. If you don't mind spending about $80, you can set up a heater system by using any 12 volt automotive/motorcycle/lawnmower battery that you can procure at walmart or any autoparts store with an inverter and a solar charger for pretty hassle free charging.
harbor freight is a goldmine for cheap stuff that you don't need to last forever -
This should do it. Be careful to keep your windows cracked as hydrogen and sulfur will probably build up as you charge the battery. Put the battery in a plastic wash basin or something in case it leaks. honestly its not really a good idea to do this, but if you're determined, thats how you can do it. Don't use a propane heater as someone suggested, propane is a hydrocarbon and when you burn it you release carbon monoxide, which is both colorless and odorless, and if you forget to put your windows down, will make you sleepy and then you'll pass out and die. its exactly how people commit suicide by piping their exhaust into their car cabin.
I also enjoy practicing in my car because, like you, I'm terrified of people hearing me sing badly. I don't mind it if im good, but I need lots of practice so the car is my favorite place. I play keyboards in a sitting position so its good for me to practice like that, even if its not hte best way to sing, logistically its the best thing for me to do right now.
But can you not just keep your car running for an hour or two? Most cars will idle for an hour and burn very little gas as there's not much load on the engine. If you spend $100 on this, that could get you around 30 gallons of gas which would probably run your car for a week on idle. plus there's the whole, you know, not dying thing.