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Whats wrong about drilling into a sheet of steel. Behind it will be 10 -20mm of polystyrene and the inner plastic moulding of your milk/egg shelf. Epoxy, self tapper, rivnut and machine screw, tucker rivet
Sliders could save you having an odd door, which you think only needs a couple of mm removing, whilst actually its likely to need the equivalent of the door thicknesses as mentioned by @Dugtheslug.
If sliders aren't going to solve the issue, is it actually worth the hassle and cost of devaluing your units to bodge this?
The sensible option would be to sell this and get one meant for the job. With a cupboard door over it, its not as if its a statement appliance as nobody's going to see it... its just a cold place for food and drink.
Just to check...it would be insane to screw into a fridge door where there is no manufactured screw hole. I know the answer but would like it validated. it seems some folk online are up for it, I'm less keen.
What it would allow me to do is attach a slider which I think might work. Then again it may destroy a £500 fridge.