Long and boring, but the proper way...
You will have to see how the cabinet is fastened. If it is screwed on the sides to the "walls" of the housing then you need to remove it before removing the "walls"
If the cabinet is screwed on the back- to the actual wall (even id it has side screws) then empty it out completely and you should be able to safely detach the wooden "wall" you want to move.
In this case you will need pad out the extra space in-between the cupboard and wooden wall you are moving with another piece of wood that will fit. You need 3" inches.
A nice way would be to pad both sides of the cabinet with 2" thick pieces of wood to give you an extra 4". Painting it white will make it look like it should be there.
Then you need to screw from inside the right side of the cabinet through the padded wood into the wood of the existing cupboard -- but not so the screw comes out the other side. You need to measure precisely the length. screw in each corner and centre. Then put the left wall back in and screw from the left side inside the cupboard the same again.
You obviously have a problem with the light switch. IF you got enough space inside you could make a square opening in the side "wall" that you are moving and put the switch in there. The wires will be behind the fridge any way - just insulate them with some flex plastic pipe for wires.
The light switch will have to get moved to something similar like this. You can mount it flush /countersink it into the wall you are moving. The biggest problem is cutting out the hole to the correct size. You can go larger but you cant shrink. So be careful and try to fit it several times.
It is not an easy undertaking- but I would not do it any other way.
If you cant be bothered ...
If you cant be bothered doing it the right way. Then put several wall bolts in the back of the cupboard in the corners and centres. Then remove the left "wall" and put your fridge in.
But this will NOT work if you have a drywall behind the fridge.
An "online" UPS is what you need. These have circuitry that converts the line voltage (which is probably bad in your case) to DC, and then recreates the AC at its output. Without this, your UPS will often switch to battery power, allowing its batteries to discharge during use. This switching of non-online UPS will also create blips in the power output that can damage the motors in the fridge.
You also need to make sure that the UPS creates a good-quality sine wave output. Cheaper UPS will create a square wave, or a very unsmooth AC waveform. Not having a smooth sine wave output will cause damage to the motors in the fridge.
The amount of power you need is a minimum of the volts multiplied by the current of your fridge, 230 times the current that it draws. Your UPS is rated in Volt-Amps (VA), which is a slightly different quantity. It has to do with the apparent power used by your load. Modern devices will have an apparent power very close to their real power, though older devices (especially motors) will require much more apparent power than their "real power", so I'd suggest over-sizing your UPS by 20-30% for this.
In addition, motors often will have a large "in-rush" current when they first start up. This can easily be double the power required once the motor is up and running. Perhaps the 1 A already factors in this inrush current, since 90 W is less than 0.5 A at 220V.
With your UPS providing 600 VA, and your fridge needing (1.0*220)=220 W, your UPS can likely supply sufficient power. The concern are if it will damage your fridge's motors, and if the battery has enough capacity to last out the voltage swings.
Best Answer
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.