Well, it felt like my problem, but I also felt she would want us to pay. She mumbled at some point 'I have to be able to explain it to my boss' (as a reaction to not let us pay by card for the low amount) so she certainly sort of showed she felt it was a problem for her too :-) But putting the work to find a solution with us.
The customer needs to make sure he/she can pay. The business needs to make sure he/she can accept pay. We both did what we needed to do, only our methods didn't match. So in my mind we had to work together to solve the shared problem.
(But well, the most important lesson is indeed to ask for card payment possibilities from now on :'))