One-Inch-Photodetector (https://github.com/aewallin/One-Inch-Photodetector) with 1 MOhm transimpedance, a larger Hamamatsu 5791 photodiode, and OPA657 op-amp.
Achieves around 4 MHz bandwidth, which should be OK for intensity monitoring etc. This will work only for quite small powers, since the maximum DC output swing is 5V, corresponding to a 5 uA photocurrent, or around 12 uW if we assume 0.4 A/W responsivity for the photodiode.
Again I needed to tweak the TIASim simulation to make the experimental data fit the model. I now suspect this is due to capacitive loading of the op-amp output. This should be possible to test by connecting the detector directly to the Spectrum Analyzer input, and then adding successively longer cables and seeing if the response changes.