Help identifying type of LC oscillator
c. Something else.

It is not an oscillator but rather a tuned tank circuit with a 10~20mA current LED indicator for Tx.

I am guessing it is a low baud rate wireless remote using AM RF available for near-field remote controls.

The tank circuit serves to suppress harmonics like an inexpensive low frequency AM modulator. 

Component ratios are critical but there is a tuned inductor. 

It's hard to tell without values, but all the reactances are critical by the gain controlled by C feedback ratios and collector-emitter impedance ratio. 

The input is a logic level perhaps a common low-frequency near-field communication frequency of a burst clock to indicate a binary level.