Problems I see:
1) How do you propose to regulate the car battery voltage down to 12v on the output of the tank circuit? You'll need a pretty special regulator there to cope with the high currents you propose. (not to mention that you'll be regulating from input voltages ranging from around 12v up to 14.5v).
2) Voltage splitter??? This is effectively a dc to dc power supply. I think the term "voltage splitter" trivialises the problem somewhat.
I have a system which works both in the house and in the car by just feeding my regular dc 2 dc supply with unregulated 13v from a car battery charger transformer and a bunch of smoothing capacitors. Works fine for me.