This is just a controlled account. We don't generally take control of accounts at other banks, usually they are brokerage or our own COD account. Usually, we only control for Assignment of Account, but I suppose it also makes sense to control for assignment of rents.
I could EASILY see us requiring or doing something like this. (including sending balances above a certain amount out). What you described above is basically how controlled accounts act... the owner of the account cannot make withdrawals without written permission of the lienholder... Sometimes there are other restrictions. Just make sure you have a control agreement in place executed by you, the other institution and the customer. If you don't already have a standard one, get your legal counsel involved.
Side note: are you going to charge for those wires out to the other bank? Just something to consider and maybe to outline in the control agreement.
As for CIP, I would guess the customer, but I am in NO way a BSA expert so don't rely on my judgment.