It is just needed to consider what is worse.
With PWM the result of the study will be "Growth speed vs. PWM duty"; with current control it will be "Growth speed vs. LED current" or worse - "Growth speed vs. Arduino PWM duty": with simple circuit you don't know how closely current matches Arduino PWM duty and light output matches current. I assume light output will follow the Arduino PWM duty "roughly linearly" while growth speed will also follow light output "roughly linearly". It will be VERY difficult to determine if the observed difference is due to electronic or biologic non-linearity.
With PWM it is simpler - you can simply verify the duty by a scope (or at least a DMM) and any observed variability is due to the bacteria. Of course it needs to be determined if it is caused by the average light output or the duty itself has some consequences.
EDIT: Tip for your next study 
With average light output fixed what happens when you change the PWM duty?