How to pick an amplifier for very fast transient signals?
The world of amplifiers is vast and complex. I have had experience with a similar problem about 2 years ago, where I wanted to measure the impulse response of a fast photodetector (also in a research setting at university).

If you are using a 10GSPS acquisition system, then you must band-limit the analog signal to at most 5GHz to avoid aliasing. It's unlikely you were going to go higher, but it's something to keep in the back of your head.
At these frequencies, reflections of transmission lines usually become important in a measuring setup, especially if you plan on preserving the pulse shapes as much as possible. Impedance matching becomes important at all stages.
As you may have seen in the datasheets on Mini-Circuits, a fixed gain block will probably give you the best performance for wide bandwidths (I have used gain blocks from Mini-Circuits myself and they worked reasonably well for my setup). However, keep in mind that they assume both a \$50\Omega\$ input and output impedance to avoid reflections over \$50\Omega\$ SMA cables. This can pose some restrictions for you sample (eg. biasing).
If you need to strictly keep the bias voltage of the solar cell constant, then a transimpedance amplifier (TIA) as close to the sample as possible is in my eyes your only option.
If you're planning to use the fixed gain block as a transimpedance amplifier, the TIA gain will then at most be \$50\Omega\$ due to the termination resistance (and this TIA gain is usually considered low). The combination of high-gain, wide-band, low-noise and stability is notoriously difficult for transimpedance amplifiers. It is not just &quot;use a fast opamp and put a resistor in the feedback loop&quot;.
It sounds to me that you're going to use a pulsed laser (I may be wrong), in which case you are in a position to average over many periods to drive down the noise. I'd say, use that to your advantage if possible. Note that averaging does not get rid of transmission line reflections, nonlinear behavior, aliasing, etc.!

At least, this is how I would summarize possible pitfalls of what I think you're trying to achieve. I've by now finished my PhD, but I remember the annoyance of trying to measure such a fast signal very clearly.