How to keep decompensated op amp in linear region?
If your amplifier behaves fine at low levels of current with the OPA846, and the problem happens only at high levels, then I believe that you would have three possibilities:

1) Reduce R1 so you have less transimpedance gain: There will be more range for the current, but you will lose resolution (amplification).

2) Tuning the gain limiting circuit (R2, C2, D1 from the first schematic in your question): If this circuit works fine with the OPA656, perhaps you can also make it work with the OPA846. Try changing R2, so that the gain control branch does not make the circuit unstable.

3) Add more compensation to the circuit by changing C1 or increasing C3. I have the impression that if the circuit works fine with the OPA656, but has problems with the OPA846, then it could be a compensation problem.

As far as I know it can be hard to think of a current limiting circuit for the photodiode since the voltage amplitudes involved are usually very low.