yes
yes, the lower drain battery will produce a lower max output
The FET cannot force a low drain cell to deliver power at a high drain rate… the FET is just an open door, that will allow as much, or as little power to the LED, as the battery can supply
lets try an example
Lets say I have a Fire Hose (the FET), and I connect it to a garden hose spigot (the low Drain battery). I then turn on the water, and I get, lets say 10 gallons per minute.
Next I move the Fire Hose to a Fire Hydrant (the High Drain Battery). Now we get 1000 gallons per minute…
Can you understand how the Fire Hose is not the component that determines the Flow Rate?.. The Hose can only deliver, as much as the Battery provides.