Oh geez! Circumstances like this are sure to be full of conditions and provisions and what ifs. The age old question of "Why did the forklift cross the road?"
A conditional point may be that a forklift is permitted to cross a road provided that it is doing so because the company has facilities on both sides of of that road and it is necessary for forklift traffic to cross in the course of daily business activities.
But simply stating that does not mean the governing authorities permit such activity carte blanche.
I have heard remarks about agriculture machinery being permitted to travel upon "rural roads" for a distance not exceeding 1,000 feet (so farmers may work discontiguous but nearby properties). Any machinery on a road in such travels must display at least a reflective tri-angular "Slow moving vehicle" sign and have lights if moving in darkness.
Here in Tennessee where heavy construction machinery and/or truck traffic must cross a public road, at least 1 flagman is required to be in place to halt traffic on the public road when construction vehicles are approaching the cross point, 2 flagmen if the public road is a divided highway type of road.

I can comment on one circumstance that I am personally familiar with. It does not involve forklifts but it does involve legally operating vehicles on a public road surface that "ARE NOT LICENSED" for highway travel.
I work for a freight company and our local facility owns a "parking lot" that is not directly connected to our main campus. This lot is used to park trailers that have been put up for sale. To access this lot we must drive 300 feet on a public road from our main entrance gate to the parking lot gate. The trailers are moved using yard tractors that are not licensed for highway travel. The only stipulation we have been told that must be adhered to is that all inter-vehicular light cords and air brake hoses must be connected when on the highway surface.
I do know that lots and lots of forklifts routinely operate on and across "back streets and alleys" to load and unload trucks/trailers.
And here in the U.S. we have the ubiquitous 3 wheeler forklifts carried on the tails of trucks/trailers to unload construction materials. Those 3 wheelers often have to operate on and across roads to deliver the goods to a site. I don't see those 3 wheelers displaying the lighting or license plates of a "street legal" vehicle.
Back to the question of crossing a road with a forklift.
Has this customer been cited/ticketed by police for crossing the road, or is this activity something the customer is wanting to commence doing on a regular basis?
If the road/highway in question is a major thoroughfare with heavy traffic it seems to me that the questions should not be so much about being able to do this "legally" but instead about "how do we do this without getting somebody injured or killed?".