Yes take the unit off the carburetor and turn the bike on with a hot fresh charged battery and in a couple minutes the plunger should extend ,,,also unplug the unit and check the voltage to make sure it's getting power,,probably should do that first
If memory serves the Lance Duke used the CN-250 Honda Helix clone engine. Yes, Partsforscooters.com or go the really big bucks and get the Keihin carb for a Honda. CFMoto used a Keihin on the Fashion which used the same engine.
You have the original? Or is it missing?,,,if it's there pull it off the bike and take the float bowl off,,,you'll probably find its gummed up with old fuel residue,,,clean the jets and you should be good to go,,,make sure you take the jets out of the body to clean them,,,,make sure they are super clean,,,all the gunk left over from evaporated fuel must be gone,,,in the bowl too