Avoiding opamp saturation - working with apd
The amplification and reaction speed of an APD depend on the current through it. It's a bad habit of the physics community to use APDs with a voltage source and a series resistor for biasing. As you have witnessed, this fails in high dynamic range applications. I suggest you change your biasing circuit to provide a constant current instead of constant voltage. 

The ringing you see is from your opamp. You are using a variant that is not unity gain stable. The -10 in the part number denotes that it's a decompensated opamp for a gain of at least 10 (Linear's opamp numbering convention). But for this kind of circuit to work, you need unity gain stability.

I suggest you have a look at Linear Appnote 92 which discusses a few circuits for APDs.