Having worked at banks in the past, and dealing with them on a daily basis in my line of work - I can give you a pretty good approximation of the response.
1) You call an 800 number from the website. You'll actually get their customer service department that isn't even in a bank. If you are lucky, they will connect you to your nearest branch (leading to #2).
2) You call your local branch. Unless you manage to get a fellow collector - they will probably have no idea of what BU means. You could tell them Brilliant Uncirculated and they still wouldn't know. Your best bet here is to just call them "new and shiny".
What will probably happen if you are able to get them to understand exactly what you want is that you will be told that they have no way of ordering anything that specific. Branches and credit unions almost always get all their coin from places like Brinks.
You will have much better luck with the main bank (usually located in the city's downtown area) or independent state banks. For example, here in Austin, we probably have 100 branches of Wells Fargo in a 25-mile radius. Every single one of those branches gets their coin from Brinks. But, the downtown location, in the highrise, deals directly with the Fed. Yes, they get stuff from Brinks too, but being a complete full-service bank, they also deal with the Fed. Independent state banks also are willing to deal directly with the Fed. You might have better luck with one of them than with the main bank downtown. They typically want your business.