Tuned circuit value switching
Your problem with power transfer is the fact that you have a huge impedance mismatch. At resonance, your LC circuit has a very high impedance, which is the source impedance to the rest of your tag. However, the power circuit represents a low load impedance, at least when the diodes are conducting.

There are a number of ways to address this. One way would be to at a second winding to your coil that has just a few turns. This would function as an impedance transformer to the tuned circuit, and would provide a better match to the rectifier.

Another way would be to select a different pair of L and C values for resonance in the first place. For a given amount of energy at a given frequency, if the L is smaller and the C larger, then the voltage will be less and the current will be greater. Again, this would present a better match to the rectifier.

In any case, you definitely want to operate the circuit at resonance. The power transfer will be much more efficient than in any off-resonance or untuned condition.