Hello,
you have an AC-coupled signal path. No matter what you apply to J17, the series capacitor will always block the DC-portion of your input signal. So the + input of the AD8237 (pin 2) will always see an AC-voltage swinging between positive and negative values. Therefore a diode in front of J17 won't help you.
For a quick test you may connect a resistor from the pin 2 of the AD8237 to V+. At the moment there is a resistor at this pin pulling the average voltage to GND. Select the value for the additional resistor so that the voltage divider from V+ to GND gives you ~100mV of average voltage at pin 2 of the AD8237 - that should keep your output always inbetween the supply rails. Alternatively applying a negative supply would of course also help.
This may solve your problem. If it does not, at least it eliminates the violation of output commond mode voltage as potential root cause of your problem.