Poiting WP installation to subdirectory

Hello,

i have one website directly in my root directory, so i installed
wordpress into a subdirectory.
I want a different site url, so i setup in the generel settings

WordPress Address (URL) https://www.URL_one/sub_dir
Site Address (URL) https://www.URL_two

Now i can access the website the way i want it. I see all the permalinks and there are no redirects or anything. So everything is good except:

When i click on the dashboard customize your site button
so i want to go to

wp-admin/customize.php

the website is not visible anymore just the menu and i get a

Session expired. Please log in again. You will not move away from this page.

message.
I think it has something to do with the bending of the URLs but i can not figure it out.

Has someone a suggestion where it might crash?

Cheers
Tim