If this is a commercial institution, you are going to have to comply with your local codes. Many will require using a certified sanitizing solution, a diluted bleach solution, or similar after cleaning.
In particular in the US, I was unable to find any direct requirements in the USDA Model Food Code, and I don't recall any from 25 years ago when I received health inspector training in one jurisdiction. That doesn't mean there are not such requirements in any given location, even if in the US.
Assuming your location permits, normally you would clean the refrigerator with a mild solution of dish soap, as you have been doing. With thick gloves, very hot water will help make the cleaning more effective. Floors, walls, and so on in a walk in would be cleaned with a solution of your general purpose detergent (back when I was in commercial service, we actually used unscented Tide, which is a very effective cleaner).
If it is possible to empty the refrigerator and turn it off for cleaning (which is not always practical in commercial environments), this doing a thorough job, and makes it easier to clean with hot water.
Update regarding moldy gaskets:
Per JES Restaurant Equipment (emphasis added):
Upon visual inspection you may find that mold or mildew is building up
on the gasket which is pretty gross and bad for the gasket. The mold
and mildew may seam harmless but over time it will deteriorate the
gasket, basically the mold and mildew will eat away at the seal.
You can combat this with a simple bleach and water solution, dilute a little Clorox in some water and use a soft cloth or sponge to apply
the solution to the moldy gasket. You will notice it start to fade
away, once you have cleaned the gasket thoroughly discard the “bleach
water”, get some clean water and a fresh clothe and wipe the gasket
down again to remove any remaining mildew and bleach solution.
In addition to being unsightly, mold on your refrigerator is almost certainly a code violation that could have drastic consequences.
Your gaskets are also probably fully replaceable, but since you are concerned with cost, this may not be your preferred option. Still, if you can, temporarily removing the gasket may facilitate cleaning it thoroughly.