Filtering current spikes in photodiode transistor selector
The "spikes" cause no issues as far as the health of the circuit.  I'm not sure of the voltage levels for V3 and V4, but you are nowhere near saturation, and your input voltage spike is less than 1 volt.

The classic approach for video is to use blanking.  If you look at your last illustration, you switched at about 5.5 uS you have the appropriate response between 12 and 13 uS, approximately 6.5 uS from when you hit the switch.  On the second pulse, you switched at 25.5 uS, and the response is appropriate between 32 and 33 uS.  There may be some other technique, but this blanking has been around as long as there have been video outputs.  An improved device or circuit might speed up the settling time, but blanking is still in wide use to prevent switching noise from getting into the video output.

So the approach is to gate the amplifier output with a 1-uSec gate that occurs 5.5 us after the switch.