Modern cars with keyless entry typically draw ~50mA@12V when "off", this power is used to power the receiver, keep ECU state for such things as learned fuel maps, blink the alarm-on indicator, etc. 45Ah capacity gets you a bit over a month to drain down the battery, while 10Ah gets you a bit over a week. This isn't good, you'd have to keep it on a battery tender if you're not driving the car for just a week. I can al…
The capacity quoted on lead-acid is also a bit deceptive. There amount of current you can draw drops pretty heavily with state-of-charge. You might have trouble starting your car even with 50% of charge left. So if Ohm doesn't work for you, you'd likely be having trouble with a lead-acid as well.
As far as the low-voltage cutout - it's definitely not trivial, but it's been done in other applications previously. I'd be a bit ashamed if you were able to come up with a solution of the cuff though, it took us a lot of work to get it working! It does involve some energy injection, but not enough to cause problems.