If you do this transfer often I recommend moving your currency to a brokerage and then doing a FX trade. You will often get close to BBG spot rate using IB.
I'd also recommend this approach. FX is a cash cow business for banks. Money changing is one thing, where the business has cash management risks and cost, but FX on a wire transfer is close to 100% profit for the institution. They trade at mid and keep that 4%.
As the parent poster commented, Interactive Brokers will give you rates close enough to mid as to not matter (e.g. the current rate to buy/sell EUR with USD is 1.2313/1.2312, plus a few dollars commission).
Sign up with a brokerage account, there are many. interactive brokers is one of the cheapest (I'm not affiliated). Then, link your two bank accounts with the brokerage. Buy/Sell you currency then transfer the funds via wire/ach...
Seems awesome for personal uses (which I expect to be doing on a fairly regular basis). I persume this might make business tax compliance a bit more complex, but at some point it may be worthwhile for me in that realm as well.
As the parent poster commented, Interactive Brokers will give you rates close enough to mid as to not matter (e.g. the current rate to buy/sell EUR with USD is 1.2313/1.2312, plus a few dollars commission).