(still learning)
What is the purpose for the smaller, gear driven flywheel? In the prototype this intended for a power-take-off belt like on a traction engine?
Yes, that is set up for increasing the rpm of the smaller belt wheel but is likely still a lower "gear" ratio than if the belt is driven from around the main flywheel, as was often done too.
Though this engine's main flywheel does look a little thin (by half) to be used for a drive belt.
A beam engine like this was often used as a power plant for a factory, driving overhead pully and belt systems. The smaller flywheel would have been the power take off for driving that kind of system.