I don't see any postulating here. Are you saying: Figure out a way to get the job done without asking permission, or become as bad as another person might be?
Yes. That is what I'm saying. I am not recommending that OP do it. Just saying this is how I think it should be done IF someone was going to do it. At least that's how I would do it, IF I were to do something. Not saying I would either.
It's simple: You are depositing at the "Commercial Teller Window" right? I sometimes go to that window (inside the "drive-thru" building at my local bank) and occasionally ask for a box of pennies. Just go there, deposit the money and let them finish the transaction. Then say, "hey, can I give you a Ten for those big coins I just dropped off?" The people at my bank would roll their eyes, but they'd DO it, at least.
Since it is only ten dollars and not a big risk, I would include an extra $10 withthe deposit and a note to the teller that I would like the dollar coins from the deposit and I would be in later in the day to pick them up. That way the deposit gets made in a timely manner and there is no discrepancy between his deposit slip and the contents of the deposit. i might lose my $10, or I would be able to pick up the coins later., or I would be given my $10 back.