You can replace the filter drier with a somewhat bigger one. Have the outlet pointing down so that it acts as a liquid line receiver making it a lot easier to get the charge level just right. While you're at it, that's a good place to add a high side service connection (between the filter drier and condenser), don't forget to remove the valve core before brazing. The low side service connection is generally easiest to connect to the stub on the compressor, where the factory filling connection is.
To find the leak, the easiest is to pressurize the system with air duster (which generally contains R152a nowadays) and then use an electronic leak detector. If you manage to find and fix the leak, add HVAC UV dye so that future leaks would be easy to find.
You'll also have to pull a vacuum before refilling. Ideally, you'll use a low vacuum gauge to confirm a good vacuum, but you can get away with just leaving the pump running for a few hours or so.
The cost of the tools probably won't be worth it to fix one fridge, but those are basically the same as what you would use on a home HVAC system.