The fridge shouldn't be on GFCI anyway. Imagine if it had tripped over the night, you woke up to make coffee, the coffeemaker didn't work and you found the GFCI tripped, and you reset it naturally. Would you register what it means when the fridge also starts up? Or would the fridge re-cool by the time you go to get food out of it, and you're none the wiser that it was ever off? Meanwhile food has spoiled.
If the fridge is downline of a GFCI, then stop using the LOAD lines on that GFCI, and fit additional GFCI receps at any other place GFCI is needed again without use of LOAD.
What to do about the refrigerator, though?
Henny Youngman had a joke, "I told the doctor 'it hurts when I do that'. The doctor said 'Well, don't do that!'"
There's a reason we advise not to put fridges on GFCI.
Fridges do that. New fridges do that. Therefore, a fridge tripping a GFCI is a nothingburger in my book.
"Oh, noes, but it's unsfae!" OK stop. Why do we have GFCIs in the first place? Because people get shocked by ungrounded (not a fridge) appliances whose electrical bits are exposed (not in the bottom back of a 300 pound machine shoved into an alcove), which get wet (not likely) and who often drop the appliance into the sink.
Absolutely none of this fits a refrigerator. They are simply not the use-case for which GFCI is intended (or able) to help. Putting one on GFCI simply makes no sense.
And indeed, AHJs see it that way - they will often exempt "Fridge/freezer only" outlets from the rule requiring GFCI for all basement and garage outlets.
Keep in mind a fridge is a safety system. You keep perishable food in it.
Chasing it anyway
If you feel strongly about running down the problem, I would start by slicing up an extension cord and separating the ground wire a good distance from the hot+neutral (bind those together). Now you can put a clamp ammeter around the ground (or hot+neutral, as normal current will cancel itself out) and look for the leakage current. I readily acknowledge that a refrigerator light is an unusual source for a ground fault, and may be worth investigating.
Normally fridge ground faults happen when the compressor motor is shut off; that causes an inductive "kick" that will rise in voltage until it is shunted to neutral or ground. Refrigerators may have devices such as a VBO to deal with that, which then fail over time.