I see what you are saying. Maybe I've inserted too many moving parts.
I do think that the refs could do the inspection in a tent on the sideline (one of those pop-up four post things that you see everywhere). The rain and wind (and sun) are variables that can be excluded, but temp seems iffy. If you just set everything at X psi in a ~70d room, they'd be playing with a noticeably different ball in a September game in AZ or TX or FL, compared to a December game in Buffalo. If a tighter ball is generally "more accurate" than a slightly deflated one, you're effectively giving a structural advantage to warm-weather teams (if you believe that, and a non-frivolous case could be made that it's real). Then again, maybe that balances out (cold weather teams would be more used to the softer ball, even if it theroetically is a less accurate missile).
So, have the inflation done at/near game temperature, and eliminate all the other moving parts
Just codify that after the balls are tested/inflated "at game temperature", any other fluctuation is understood as part of the game. (I mean, it's still football...the refs are suposed to be taking the time to check the balls anyway, and the refs can go outside for a while, and a simple fabric tent would shelter them from wind and rain, while providing pretty close to "game temp" conditions)