Can't send with my personalized iCloud+ adress in Mail app

I hope someone can help me as the apple support and my domain provider can't :,-(

I created a personalized e-mail address in icloud+ to a domain I already owned. I got instructions form icloud on how to change the DNS of my domain. That all worked perfectly fine and I can send and receive e-mails now from my iCloud Mail in my Safari browser. But in my Mail app, I can only receive, but not send, with this personalized e-mail adress.

The apple support says it's my domain provider's problem, the support of my domain provider says it has to be a problem of iCloud. So I'm completely stuck and don't know how to solve it.

It must be something with the SMTP settings I think. The pre-settings were:

user-name = [my personalized e-mail adress]

password = [empty - it says "optional", so that should be ok]

host name = smtp.mail.me.com

I also tried changing the user-name and password to my icloud login information, but that didn't help. The error message says "Client host rejected: Access denied". To make sure it's not a problem of my internet provider I switched of my WLAN and tried it with a hotspot connected to my phone provider. Same result.

Any ideas what else I could try?