The LP regulator is not dependent on atmospheric pressure to operate.
It operates off spring pressure on a diaphragm and inlet pressure on other side of diaphragm.
The functionality of the appliances is dependent on air/fuel ratio which can be affected by altitude.
Fridge & furnaces seem to be less affected due to fixed metering of gas (small flame on fridge) and blower supplying combustion air on furnace.
Water heaters tend to be more affected due to distance from orifice and where flame occurs but they have adjustable air shutters that can be used to correct mixture
I have yet had to do anything to any of them when at high altitudes.....CO/UT/WY/MT/ID etc.