Bitcoin is actually more expensive than SEPA transfers or girocard payments, and slower to resolve (and just as impossible to reverse as SEPA transfers).
Which bank is this? Since 2014, SEPA-ICT has been introduced, and since this year, every bank should support it.
Normal SEPA transfers are mandated to be free and < 24 hours, and SEPA-ICT transfers are mandated to be < 1.5€ and < 15 seconds, but most banks implement those for free.
As you can freely choose any bank in the EU, you might want to consider German Commerzbank, which implements free < 15 seconds transfers to any other SEPA bank (except for student accounts, those have to pay 1.50€ for those transactions).