"Wise" (previously known as Transferwise) is an excellent & cost effective method to move money internationally whether you are changing denominations or not. Works just fine for six figure sums too. I can't recommend it highly enough.
I have no connection to "Wise" other than as a satisfied customer for 5+ years.
EUR100,000 to GBP modeled out below.
5k limit - Urrgh, that's brutal in this day & age.
Give the branch notice that you are planning to attend in person to make a large outbound transfer (name the amount) & request an official appointment & request that the Branch employee is both capable & authorised to make such a transaction.