I did find a complement for MPSH10. Unsurprisingly it is MPSH11. Unfortunately, besides having limited availability, it can't sustain more than 50mA of continuous collector current. The second mirror amp circuit looks like it could put at least 100mA continuous through Q262, so no go.
They're probably wanting small collector-base and/or collector-emitter capacitance to keep the trigger output fast. A non-optimal substitute will likely just result in a slower trigger output.
The only thing I'm coming up with so far is 2N3209. And availability is not good. Hopefully your 1853-0357 order will come through.
If all else fails, you could try a pair of common 2N3906 and see what the output looks like. It won't hurt the unit if it doesn't work.
My trigger output into 50ohms has a T_rise of 4.0ns and T_fall of 3.3ns, for reference.
Edit: Not a current mirror, rather a differential amp.