It's how the banking system and the financial markets authority work in Europe. With the European laws, you're not so free financially speaking. you can't make whatever you want with your money.
To give you an example, in my country, your account is flagged once you hit 5000€ worth of transactions within 30 days. So, the bank has to abide by the law and report your account to the authority govt. then the authority will scrutinize the last transaction to see what you're doing with your money.
Dirty coins or not doesn't matter since you don't transfer coins to your bank account, but you trade for fiat and only then transfer outside.
An exchange or a web wallet can also ask you from where the coins come from and what is your activity
To give you an example, in my country, your account is flagged once you hit 5000€ worth of transactions within 30 days. So, the bank has to abide by the law and report your account to the authority govt. then the authority will scrutinize the last transaction to see what you're doing with your money.
Dirty coins or not doesn't matter since you don't transfer coins to your bank account, but you trade for fiat and only then transfer outside.
An exchange or a web wallet can also ask you from where the coins come from and what is your activity
So then what? If you are transferring money from exchange account that is in your name to your bank account that is in your name, what is the problem? If you then transfer it from that account in your name to another of your accounts elsewhere that is in your name, what is the problem then?
It is your money and you took it out of coins to put back into your bank account/s.
Would they require proof in this circumstance and if so, what kind? Probably, best to keep it under the 10k limit?
.Probably they'll ask you for another document if your id does't have an address or etc. In my case only my id was required in 15 min I was verified with 25k euro limit, 2.5k per transaction and 5k per day. 
Revolut and Monese are the backup plan for top ups. No need loans nor anything that doesn't accept sepa from kraken. With iCard you got 3 virtual cards a contactless icard, virtual visa and virtual mastercard. All got daily limits 5k eur with 2.5k eur per single transaction as the first got 15k eur weekly or monthly limit so 25k at all limits per month. Im waiting for my plastic one as 4th card in the app and I expect additional limits there. 