Your NGINX rule captures the incoming traffic to the /register location, but it doesn’t redirect it anywhere. To be able to access the /register page, you should add something like try_files $uri @proxy_to_lms_app; at the end to have your request passed forward.
While this is going to work, you can consider additional layer of blocking the registration page, because this one can be bypassed like this:
Enter /login page.
Click “Create an Account.”.
If you want to disable this, you can set ALLOW_PUBLIC_ACCOUNT_CREATION feature to false to disable the mentioned link and Register button, so you will be able to access the /register page only via its URL.