I built a small portable power system for camping/emergency use and wanted the same thing. I ended up going with a latching relay as it only requires power to turn it on or off, then it latches in that position. You connect the positive, then there are two grounds that can be completed by a transistor to turn it on or off depending on what ground you apply power to. This is a "dual coil" relay, there are some single coil ones but you have to reverse the polarity, which would be a bit more tricky to do with a MCU. I think a H bridge is what you need. Dual coil is just simpler to setup.
In my case the microcontroller has other tasks such as controlling LCD display (voltage and other messages). I can then connect a laptop to set different parameters such as cut off voltage and calibrate voltage/amps readout. I can't seem to get the amp readout to be accurate as it jumps around too much, but that's another story. If I was to do it over again I would have used a high side current sense IC instead of shunt/op amp like I did.
11 volts is a decent voltage set point to go with. You could do 11.5 to play it safer and extend life of battery. 12.6 is way too high, you'll barely get any use out of it. That's practically a full charge sitting idle with no load. It will go lower with a load. I have seen some 48v systems that stick around 50 volts for hours though with a load on them (that would be like 12.5v on a 12v system) but batteries vary.
Idealy whatever system you design you should not hard code stuff like shut off value, so you can field adjust it as needed. That's what I did for my camp solar system, I can plug a laptop and there is a text based interface to setup different parameters such as voltage calibration, shut off, shut off timer, etc... Was fun to code.