My garage is attached to my house and adjacent to my home office so I was often in range of my car when I was working. The battery being crushed isn't an inherent function of mobile key; it's likely a function of the background permissions including always on location by default, which runs your GPS constantly, which torches your battery. When I turned that off and set it to only allow location access while using the app, my Lucid app battery drain (visible in the Battery settings in iOS) went from 38-40% of my battery drain down to 1-3%. It is an extraordinary difference, and the mobile key works better than it did before.
The delay on my mobile key unlock has also been substantially improved. Now it almost always is unlocked by the time I get to the car door. These things don't feel like they should be connected (especially since the app whined at me that mobile key wouldn't work without always on background location permissions, which was flatly false) but in my experience they were. YMMV. I'd suggest you try it.