Cash must be declared even within Schengen countries.
No fee is due on declared cash. That's your own money! You must call a customs officer at the departure airport and ask to declare money for >10000€ in cash.
The reason is fighting money laundering and tax evasion. The declaration form contains your personal identification data, and your EU individual tax code. You are free to move your own money across countries as soon as you declare them.
Tax authorities will collect declarations and may use them
against you in case of income verification or investigation. Basically you are leaving a trail that you own large amounts of money and that you are moving that money in cash (untraceable) from country A to B.
Tax authorities can already monitor traceable payment methods, and they do use big data computing to find anomalies and investigate potential tax evaders.
Conclusion: it is perfectly legal and allowed to bring cash abroad, but you must declare it.
Source: I work in a financial RegTech company and I am familiar with these kind of operations (e.g. compare income, property and expenditures after data submitted by financial operators)
I haven't answered for the part about airlines. You may want to check directly with them with their travel conditions. I would suggest to (passively) read their ToS than calling them. We are still speaking about a lot of money.
Anyway it's the security attendants who will see all that cash in the xray machine for the first time. Airlines don't check luggage themselves.