Why is the current in the resonant load in my H-bridge only sinusoidal for low current amplitudes?
The vertical slope of the BH loop is proportional to inductance and higher impedance.  Notice that have DC voltage offset thru the coil, where the average DC current rises by integration over time until it reaches the B limit or Bsat value and then L inductances drops to zero if very flat peaks and your AC waveform is now driving near zero ohms so high current peaks. Make sure the drive is balanced as the name suggests in &quot;Alternating Current&quot;

Something is obviously wrong with the design of the coil drive with DC offset or burden load resistor value.  In order to detect 1000A peaks there must be no DC thru the windings.
Ensure there is no DC and load Burden Resistor is correctly chosen to ensure it is properly loaded.
Also read about Remanence and degaussing due to abrupt current shutoff in a transformer  which also causes saturation. This is done in the lab with a Variac.
BTW your signal integrity is very poor on &quot;The Good&quot; side. This needs significant review and improvement with more details than you have offered.