That circuit is a linear one so there is no possibility of modulation unless you want to include some strange behaviour inside the battery (not altogether possible but . . . ). As you haven't included any switches then I am not sure how you want this circuit to do AM.So this would work and oscillate until the battery lost it's charge? Is there a circuit like this I can read on. That's the main problem, I don't have a resource for this type of setup.
If the battery is a true voltage source then it has no internal resistance and it can dissipate no energy and the solution is indeterminate. If, as you should do, you introduce some resistance in the circuit you can combine the battery resistance, wire resistance etc. as a small series resistance on the output connection of the battery and you can consider it as a normal RLC circuit. (It is always best to reduce complicated networks as much as possible before analysis)
No net charge can leave the battery after the first cycle of the resonance because the Capacitors are open circuits. It will not 'go flat'. There will be an oscillation with a natural frequency of 1/2π√(LC) where C is the equivalent Capacitance to the two Capacitances in series and the L is the equivalent to the network of self and mutual inductances (I assume that L1 and L2 are coupled in some way?). The oscillations will die down exponentially and the final voltage across the Series LC part of the circuit will be equal to the battery voltage. The only energy lost will be the initial energy that flowed into the circuit at switch on (CV2/2) and the number of cycles to drop the energy to 1/e will be the Q of the circuit..