BJT with fast turn off characteristics
The transistor turns on and off fine (red arrows) but as you noticed there's something that looks like a RC discharge curve (highlighted in green).
This is due to the LED*'s internal capacitance. When the transistor is on, the LED's capacitance is charged to Vf. When the transistor turns off, the LED capacitance will discharge, rather slowly, through the LED itself which creates exactly the type of curve shown on this scope shot. This means the light output will have the same decay instead of turning off quickly.
(* I call it a LED, maybe it's a laser diode, but you get the idea)
The way to solve this is to discharge the LED capacitance when turning it off. That's why I suggested driving it from a 74ACT logic gate, which has push-pull output and 3.3V compatible input levels. That's also what the recommended schematic in the datasheet does.
However you can't do that since you don't want to change the PCB and you probably don't have enough room for a SOIC14 chip, and I couldn't find any 74ACT in single gate SOT-23.
You could simply stick a resistor in parallel with the LED to speed up the internal capacitance discharge when it turns off. Start with doing that and see if it fixes your RC decay. If it does then my hunch is correct and the culprit is the capacitance.
If this is the case then you'll need a push pull driver, either implemented with two BJTs or logic gates.