For me, it depends on the bank. I don't dump anywhere near as much as you, but I try to limit myself to $1000 - $1500 per bank. I search around $5k in coin a week. Coin lock bags are awesome, but they aren't really an option for me right now. I have two bag options: 1. a bank that takes two weeks to credit the account 2. a bank that is serviced by my supply armored car company. Neither of these options really works for me.
All of my dumps go in counters. The amount and type of coin depends on the bank and type of counter. My least used dump bank is TD because the half bags fill at $100 increments. I mainly only dump at TD when it's quarters and dimes and I'll take up to $1250 in dimes and quarters in a trip to them. I also have a credit union that I've started dumping at. I'll take up to $1K in halves at a time there because the bag changes aren't frequent and they are serviced by the car company that I love dumping on.
My main half dumping bank is a small local chain of banks (~10 branches). Their coin counters are super fast, they NEVER have to change a bag, and they aren't serviced by the company I get my supply from. I usually take $1250-$1500 in halves to three of their branches each week.
With all that being said, I have never had anyone cut me off. There are certain branches that I end up in every week, but for the most part there are enough branches of each of my dump banks where I can rotate them so that I'm only in most of them every other week at the most. I think long term that this methodology is going to allow me to keep from getting cut off for the longest amount of time. I've been thinking of upping my half volume to 10-20 boxes per week this summer, but honestly supply and dump banks aren't what is keeping me from doing it. I just don't think I have enough time to pickup, search, and dump that much coin.
My dumping strategy this week is different: Last weekend I went into a TD and asked to buy a box of dimes in an area far from my house. The teller was downright nasty and said that dimes were only for business account holders. Like really nasty. I'm a nice guy, so since she is obviously experiencing a shortage of coins I'm going to help her out this saturday during lunch time by bringing her $4k in halves and $5-6 k in dimes and quarters.