I would use high temp disc brake caliper lube available at any good autoparts store. I work on school buses I have done wedge brakes and standard roller cam style . It should work once you clean everything.

I use blue high temp brake grease on my M1A and other firearm reciprocating parts.
Been sitting on my side table next to my recliner for 15 years.
Kids call it Ancient Grease. Hahaha

Same here. I'd even go one further and go with a synthetic like this

https://www.amazon.com/Permatex-24110-Ultra-Brake-Caliper/dp/B000HBNV6W


Non-Melting, corrosion resistant, Hi-Temp, etc. Designed for the heat of Calipers and pads so wedge brakes are a piece of cake for this.

It goes without saying that everything should be well cleaned before reassembly even if you are going to use the OE product. Failure to clean brake mechanisms before re-lubing is a major cause of "bad language" and other problems

I looked up the spec sheets for all those caliper greases and they are NLGI #2

I'm afraid they would be too thick to let the pawls work in the adjusters

Especially a non-synthetic base oil mid-winter.

I was not understanding what you said then. If you are talking about the adjuster your grease of choice out of your grease gun should work. assembling brake components is where I use disk brake lube.

I think it's more complicated than that

It needs to be the right grease

I found a cross reference to an industrial grease that I can order from McMaster-Carr