An online service like currencyfair.com, transferwise.com or worldfirst.com (no connection to any, although I've used currencyfair) will almost certainly beat your high street bank both on the exchange rate and the fees, and their websites make it very easy to get precise quotes.
You can then decide if it's worth the time/hassle of creating a free account just for the purpose of a once-off transaction. If you have scans of the necessary documentation and time to take a 'phone call for verification purposes, it's not too arduous. You might save several hundred euro.