Transient Concerns in Wireless Power Transfer

I've encountered a curious situation. I have a device that is continuously wirelessly powered by an inductive link. The Rx coil is attached to a resonant cap, a matching network, a rectifier, and then a voltage regulator. However, I had an issue with being able to supply enough power instantaneously for a micro-controller on the device. This problem persisted even with increased output power. It was resolved, however, simply by increasing the inductance of the receive coil (note, the overall efficiency of the inductive link is not dramatically effected and the resonant cap and matching network were updated for the new coil). This seems to have solved my problems entirely.

My theory is: Even if two LC resonators have equivalent Q and w0, the LC resonator that has a greater L is able to source a greater instantaneous current.

My questions are thus:
1) Has anyone encountered a similar transient problem in wireless power applications?
2) If I have two LC resonators, both resonating at the same frequency but one having a greater L than the other, does it make sense I would be able to draw current instantaneously from one faster than the other?

All of my experience in WPT has been in maximizing power transfer efficiency, which for me is exclusively frequency domain, steady state analysis. The idea of a new set of design constraints based on transient power delivery requirements is pretty cool to me, and I would love to hear if someone else has bumped into this issue. Or if someone has reason to think my theory is completely wrong, I would be interested in that as well.