I think you should add any capacitance on the Primary side, not on the secondary. I think this will give you lower losses. You can model this in some kind of SPICE program.
I think there is no particular disadvantage in adding the capacitor. It will reduce the resonant frequency, so you might need fewer turns. Is there a particular reason why you wanted the frequency around 10MHz? You could use the ISM band at 13.56MHz, or someone else might know about a lower frequency which can also be used for ISM proposes. I think under
150kHz the rules are also less strict, though I don't know the details.Ok, a LC oscillator will do that.
It is pretty easy to make it oscillate, and only 4 components are really needed.
Ok, but it is an ugly way of passing. Sort of like making the percentage of pollution in your drain pipe below the legal maximum by leaving the tap running all the time.