How do I set the flow rate on my Ender 3 Pro and have it stay at that percentage?
Changing the flow rate during a print can not be saved. There simply is no way. It is usually meant to be a fix with filament inconsistencies or to look for the right extrusion factor for a new filament batch.
Slicer
The only way to consistently increase the flow rate would be  to alter the flow rate in your slicer to what you have found to work best for each machine, probably using separate profiles. This will up the rate for every subsequently sliced print. Note though that this 108 % increased extrusion is converted extrusion factors that are simply numerical and 1.08 times the normal in the g-code. These numerical values will be taken as 100 % by the printer - and since it requires extra work to slice the gode for different profiles it is not the optimal solution.
As you elaborated though, this is not a doable thing, so let's look further.
Source hunt &amp; Workaround
Since only one printer is showing underextrusion while the others do not, it is time to check the hard- and firmware:

underextrusion can be caused by a defective extruder assembly or a damaged or blocked nozzle.
if a machine has consistent underextrusion, its steps/mm in the firmware might be off. This could be altered and stored in the EEPROM. Since this could be a machine unique setting, here would be your point of attack to increase the extrusion of just one machine while using the identical G-code to all other machines.

Note that the standard firmware of the Ender-3 in 2019 did not contain Thermal Runaway Protection (What is Thermal Runaway Protection?) and should be upgraded because of this anyway. You have to flash a bootloader too, so in the process of doing the upgradeability and safety-upgrade to all the machines, you could store the altered steps/mm to each machine individually so they get consistent output.