Can you insulate the ducting more to keep the air in the ducts warmer?I don't know much about home furnaces, but I would assume they are similar in many ways. Does the Furnace in question with the "Forced air" have a fan? I think home furnaces running with gas would have a a built in safety feature since your dealing with natural gas. SO the schematic would say something like, once forced fan on turn on primary for the fuel solenoid or gas solenoid ect. Otherwise the unit would melt itself. and prolly a couple of thermostats to detect low and high heat points to shut off the primary as well.
If the one fan in question runs at the same voltage as your duct fan, then couldn't you hook it up in parallel with those connections, Then the fan would come on when the heater is running.
I would just wire a new connection for the fan, possibly hook up a rheostat dial or some the like. tell ya honestly, make sure all the vents have ample room to draw in air, and make sure the ducts are all cleaned seams are all sealed , no holes ect...as well just for precaution.
Marty