It’s some kind of fairy tale that the prices offered online don’t also apply if you go there in person. If the bank offers 100,000 euros for 1% at a 50% loan-to-value ratio, then you get this 1% (or better) – or nothing at all if the bank refuses. But what arguments should the bank have to worsen its offer? Loan-to-value? That was clearly specified beforehand?
The bank will not change the fundamental conditions, but the definition of "50%" is up to the financing bank.
and about the interest rate: more important than the last few hundredths behind the decimal point is the general handling of the process.
That’s not an unimportant point, but it also depends on whether we are really talking about the per mille range here. 1.9% versus 1.88% – no question. But when comparing 2.1% to 1.9%, I would start to think twice, that’s around 10,000 euros over 270k in 20 years; for that, I would personally put in some effort and, if necessary, also accept a few hundred euros in reminder fees.