They would like a lock that has a built-in non-resetable mechanical counter. Every time the lock is unlocked the counter should increment. They want each person to record the number when entering the room.
So far I have been unable to find such a thing, although it seems like a very useful device.
Does such a thing exist?
There are non-resetable electromechanical counters. If you have an access card reader activating an electric strike in the door jamb, then the pulse to the striker could also go the the counter. If you don't use access cards, then a push-button could be used instead.
With the access reader and electric strike you could simply get a log of all card swipes and avoid the need to write down the numbers. Plus you would get time of day for each access.
So, you have to look for those systems that acknowledge this fact, and work within these rules. They should be certified to resist certain types of attacks for a certain amount of time, and there are national and international standards for all sorts of these things.
The goal should be to slow someone down long enough that other measures can also take effect, like security guards walking around, or someone watching a camera, or whatever.
Look for mechanisms that cite the standards that they adhere to, and then go looking for those standards. Look for related standards, or higher level versions of those standards. Look for products that adhere to those standards, and then rinse and repeat.
Make sure to set yourself a time and money budget before you do this process, so that you don’t waste too much time going down too many ratholes.
I think, the lock with the counter became inoperable instead of rolling over and had to be replaced.
It might have been for the US postal service.