Even if it's a problem with off-the-shelf stuff, I imagine a car-manufacturer could easily get something all nice and tiny and special-purpose.
Airtags transmit much more frequently than car remotes, use similar batteries, and yet do proper security.
I've become a big believer in leveraging some security features of the physical world, as it seems it's been long enough that everyone's forgetting Therac-25-style problems. (Or, perhaps more accurately, nobody cares because they aren't liable.)
Modern keyfobs actually detect motion and if they are motionless for a while, they stop transmitting the signal to both save battery and prevent such attacks.
For old keyfobs, you can get a battery sleeve with integrated motion sensor which does the same (cuts power when fob is not in motion for a while).
Alternatively, some cars let you disable the feature and just use the keyfob as you would use an older one - then you habe to push the button anytime you want to unlock the car.