Day tripper thank you. You get the question. So from room to fridge I don't have to change anything to keep same carb level.
If you disconnect the gas line from the keg, then no, you don't need to change a thing while the beer chills down to serving temperature. Then, as noted, when you reconnect your gas supply, you'll need to change the CO2 pressure to the appropriate level to keep the carbonation level consistent.
Otoh, if you were to leave your CO2 supply connected to the keg through the chilling process, then you would definitely need to adjust the regulator downwards to track the falling temperature, lest risk some degree of over-carbonation.
Not only would that be a minor pita, if you happened to have over-filled the keg such that the gas In dip tube was submerged in beer, and you didn't have an anti-backflow valve (aka "check valve") on the gas line, you would risk having beer back up to the regulator when you turn the gas pressure down.
So the easy, essentially risk-free way to go is to to carb up, disconnect the gas line, chill down, then reconnect the gas at the lower pressure...
Cheers!