Hi @frantisek.pernicka,
Thanks for following up! I had another look at this with a colleague of mine, and we’re wondering if it might be the same root cause of an issue we’ve seen recently regarding a change since our implementation. This might be a shot in the dark so please let me know how you go. 
This was regarding the fact that password doesn’t seem to work now against Jira cloud (gave us the same auth failure). Access Token worked fine, so I’m thinking this might be the missing piece.
Quoting Atlassian’s doc page: “Basic authentication uses an API token to authenticate the client. The token is generated from a user’s Atlassian Account, encoded, then added to the header for requests to the API.”
Quoting the section in the docs that I believe is vital here in the issue you’re facing (within step three of this linked doc section):
“The password should be an API Token, rather than an actual password. You can create an API token from an Atlassian account in the ‘Security’ area.”
Can you confirm if you’re using an API Token here?
I hope this helps get you going, and please let me know how you get on.
Best regards,
Kenny