{(This might sound silly)I was wandering: is there a type of "locking" relay where you provide current at the beginning only to switch it close and then after that it "locks" into position by somekind of mechanism which allows it not to drain any more current to operate the electromagnet?}
Yes, such a relay is called a 'latching relay' and they come in two coil or single coil versions. In the two coil version you pulse either the set coil or the reset coil and relay moves to that position and is magnetically retained there so no continuous coil current is required. In the single coil version you apply one polarity of voltage or the opposite to set or reset the relay and it will retain last command with no continuous coil current being required also.
On your lipo voltage levels, most recommend that +4.2 vdc is fully charged and 3.0 vdc is a commonly used cut-off 0% voltage recommendation. If you use 3.6 vdc as your cut-off value you will be leaving quite a bit of battery capacity unavailable for your use thus lowering the effective mAH rating of your battery pack.
Lefty