idratherbefishing,
The vent is not only for the diaphragm balance for the regulator function, but also fulfills requirement of fire code NFPA 58 for over-pressure relief between regulator and burner. There is an internal relief valve on the diaphagm head that would flow out through the vent opening.
You can locate the second stage regulator indoors, and find a way to pipe the vent to outside - maybe same route as the appliance exhaust vent to the roof or out thru side wall.
The typical location for a second stage regulator would be at the gas line entry point to the buiding - either outside on the wall or inside the wall with the vent piped to outside the building wall with an end fitting of bug screen with rain + ice shield.
It sounds like the appliance is 130 feet distant inside a building, and there will be a trade off between the cost of vent piping versus cost of larger size piping in place of the 5/8" copper.
Is the appliance an unvented space heater with oxygen sensor shutdown? Otherwise, how does the appliance take combustion exhaust to outside?