I see no need to check a deer at the kill sight, that is why their is the 48 hour time period. Kill the deer, drop off at processor, record kill next day while on stand. I always have check number emailed. Don't look at number till back at work in the office, which could be up to a week later. Don't plan on changing those ways, don't see a need to change them(well within the requirements), Wardens will have to get over it. The number is dated and recorded in the system, should be up to the game officer to look that up if they feel a need to, not the processor. They are simply using the processor/taxidermist, and penalizing them, to do the officers own d*^% job.
Processor/taxidermist can have a small printer or camera to take a photo/copy of hunting license. Wardens can use this to look up the game check number and date from there, processor doesn't have to deal with it.