Ground the fridge!
Grounding the fridge will not defeat the purpose of the isolation transformer, but rather complement it.
You may be operating on a misconception that GFCI devices care about ground. They do not. They are not connected to ground in any way and have no idea what is going on with ground. Actually, what grounds do to GFCIs is make ground-fault current more likely to flow, which trips them obviously. The isolation transformer will prevent that.
The reason to ground the fridge is to assure the fridge chassis and the sink, light switch cover screws and other grounded things remain at the same voltage. It will also assure a GFCI trip if the isolation transformer itself leaks, something you want for safety.
An isolation transformer is a good solution, though
In this configuration, the GFCI will only "see" the isolation transformer primary. Currents will be equal in the transformer, where else can it go?
I never believed any of that "Harmonics in the AC motors ingest some of the electrons, causing false trips" nonsense, but if that were so, the electron losses would be on the isolation transformer's secondary, and the GFCI would not see them.
Fridges shouldn't be on GFCI in the first place
We have never found anything in Code that requires refrigerators be on GFCI, except for ones in garages or basements only befause garages and basements need GFCI, not the fridge. Even there, authorities tend to be warm to the idea of dedicated non-GFCI circuits for fridges.
Look at the reasons for GFCI, they simply do not apply to a grounded refrigerator. However, fridges are life-safety devices; if their GFCI trips and someone resets it, they may not realize the fridge was off and the food has spoilt. Then the aide comes in to feed grandmother, and aides have an ethic about not eating your food, so they never check.
Old refrigerators should go anyway
Because new fridges are much more efficient. It may seem like a cost savings to run an old fridge, but that is only a lack of information on your part. You are not seeing the cost of the electricity for the fridge. If you put a Kill-a-Watt or similar meter, you would get a baseline. If you then bought a brand new fridge and did the same test, you would find the new fridge is saving you $10/month or more. Since this test is not practical for you, well, you never find out!
The upshot is a new, efficient fridge will pay for itself in a few years with the power savings, and then the savings are pure profit.
Some electricity companies are either subsidizing or simply giving away new fridges to customers with obsolete fridges. Replacing 300MW of fridges with 100MW of fridges is cheaper than building a 200MW power plant. The cost is covered by ratepayers on a per-kilowatt-hour basis, especially the ones who are still running their inefficient fridge.