Ok,did some math by looking at drill but numbers, their associated diameters, and reduction in surface area.
Going from a #56 diameter orifice hole to a smaller #57 diameter orifice hole after soldering it shut and re drilling reduces the area of the "hole" by approximately 15%. If this is enough depends on many variables but should be a close starting point. Many stock fridges run just as fine at high altitude without changing orifice size. Its possible that your fridge just needs a minor tweak. If this doesn't fix starting or running, then by all means try the next smaller size thereafter.
As a pilot, we use a general rule of thumb for 3% less power output of an engine due to thinner air with every 1000 foot gain of altitude...a simple approximate rule of thumb that doesn't take into account air temperature/density. If you are camping up to 8,000 feet, going one drill bit size smaller in orifice diameter may not be enough and perhaps 2 drill bit sizes would be better.
I'll let you decide what diameter to use if you are concerned. I checked my Dometics and Norcolds at the top of a 12,127 foot pass and both started up and run fine at those altitude and BOTH WERE STOCK. I think you'll be ok with one diameter smaller size but going 2 smaller sizes if you camping at 8500 ft and higher wouldn't hurt.
Buying spare orifices is not very cheap but I'd rather spend 60 bucks on 2 orifices to have easy to replace options at higher altitudes. I'd have stock size for sea-level to 4000 feet, one size smaller orifice by soldering and drilling for 4000-8000 feet, and orifice soldered and drilled 2 sizes smaller for 7000 to 1200 feet.