Here is some of the promised data. As mentioned, changing diode current and crystal temperature didn't have much effect on the frequency noise spectrum; but the spectrum itself does seem too high for our needs. 

At each temperature, we started measuring the spectrum at 1.8A, and stepped the current up, hoping to reach 2.0 A.

At 47.5 C, we were able to scan the current from 1.8 to 2.0 A without much problem. At 49.0C, the laser mode would hop away above 1.95A. At 50.4C it would hop away above 1.85A. The spectra were not seen to change when physically disconnecting the PZT actuation BNC from the rear of the laser. 

The flattening out at the upper end is likely due to the SR560 output noise. I foolishly neglected to record the output spectrum of it, but with the marconi external modulation set to 3.2MHz/V, the few Hz/rtHz above 20k translates to a signal on the order of uV/rtHz, which seems reasonable. 

Data and code attached.