This is a generic response. If you think there may be nuances unique to Florida, please re-post in that forum.
Safe deposit procedures are like the net below the high wire. You really don't need all of those strings, but it may make a big difference which ones you choose to cut if something happens down the road. What all of your procedures come down to is that if a renter alleges that there has been a mysterious disappearance of his property could you convince the judge that it was simply impossible that anyone but the renter had access to the box?
Obviously, a time and date stamp would be more reliable than the same information handwritten by the custodian.
If a system was designed to facilitate changing locks; i.e. it is easily done by a bank employee, it was intended that the locks be changed every time a box is surrendered. Failing to do so would actually make your bank look sloppy in the event of a mysterious disappearance. If the mechanisms require the services of a locksmith to change them, then rotating locks with every surrendered box is generally thought to be cost prohibitive as long as both keys were returned.
Whatever your decisions may be, it is prudent to homogenize your procedures across all offices.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.