Function generator changes output amplitude when connected to resonant inductive coupling circuit
At a large distance, the loading on the signal generator is just a series tuned circuit (L1 and C1) having a net impedance that is solely R1. I say this because I'm assuming you have it tuned to maximize resonance effects.

The signal generator is never going to be able to maintain full p-p into a 10 ohm load but how come it rises when you bring the two coils together....

In simple terms, when the two coils are brought together, L1 starts to "adopt" the impedance of the secondary tuned circuit made of L2 and C2 and, because this is a parallel tuned circuit, "L1" begins to rise in impedance as the two are brought together. Thus the output voltage of the sig gen rises. That's the simplistic but fairly numerically accurate version if L1 and L2 are similar in value.

If you look a bit deeper you have to recognize that the induced voltage in L2 causes L2 and C2 (if tuned correctly) to act as a series resonant circuit - this is because induced volages are in series with coil L2 and not across the terminals so, L2 and C2 start to series resonate and the voltage across L2 (as well as C2) rise as the distance reduces.

And, because L1 and L2 are coupled like a transformer (ignoring the not insubstantial leakage flux), the voltage across L2 is tending to force itself across L1 and detuning the primary tuned circuit. As the primary series tuned circuit detunes the impedance is presents to the sig gen rises and the voltaage at the terminals of the sig gen rises.

For such a simple looking circuit there are some significant subtleties.