I cannot see an error in the developer window, but I am not an expert.
You need to review the Console tab for JS errors. JS errors will be in red. It sounds exactly as if the Stripe JS is failing or your site. If still unsure how to review please PM backend super administrator login credentials and will take a look (see signature for link to PM me).
By the way: the other gateway 'QuickPay' runs without any problems in my template. But I want to change it because Stripe has 'SEPA Lastschrift' in its options.
QuickPay isn't Stripe. No two gateway implementations are alike.