Virtually, any current digital scope would be fast enough for this job. Pressure buildup in the chamber will be in the low millisecond, maybe several hundreds microseconds range. Any entry-level DSO today will record a sample every nanosecond (1GSa/s) or faster.
The more problematic item in the signal processing chain is the strain gauge amplifier since the typical output voltage of the strain gauge is very low. Hence, often lock-in amplifiers with heavy low-pass filtering are applied.
Since the measurement itself will be very short and since the output signal of the strain gauge is ratiometric to the excitation voltage, you may consider pulsing the excitation voltage to a much higher value than the strain gauge would be able to withstand continuously, thus boosting the output level by that factor, and using a faster, low noise operational amplifier for signal amplification. Tests would be required to verify that this approach doesn't affect accuracy due to strain gauge heating.
For triggering the measurement chain, you could use either an optical interrupter type sensor to detect the cock approaching the firing pin (still many milliseconds to go...) or an accelerometer system that senses the vibration when the cock hits the firing pin (a few milliseconds before the shot will break).
...just some food for thought...