The electronics of newer furnaces are, by design, very sensitive to any voltage between the grounded conductor (neutral wire, white wire) and the equipment ground (green wire or conduit).
So the wiring of the generator and transfer switch together can affect that.
In addition, some electronic boards are very sensitive to noise or voltage distortion on the input power, and that will vary with generator design.
Another possible problem is a generator with some "energy saving" mode that only delivers full voltage when a load above a minimum value is sensed. If there are no other constant loads, this will not let your controller work.
IMHO, the starting point would be accurate voltage measurements from line to neutral to ground (three measurements in all) with each generator.
If the results are identical, then it will get trickier and may require more specialized tools and knowledge, not necessarily possessed by a typical HVAC tech or electrician. You will need to find someone willing to commit to solving your problem.
It may be as simple as a wiring error or installing a noise filter. Or not....
