Health Check with Authentication when Initial URL Redirects

On an LTM with 11.4.1 code:

I've been tasked with creating an HTTP health check that requires authentication. But there's a catch:

Initial call will be like http://1.1.1.1/config/ExtendedHealthcheck.hc

However, going to that initial page redirects to http://1.1.1.1/config/login.jsp which is where the login is presented. Oh, and that page presents a JSESSSIONID cookie as well. After successful login to the page you get to /config/ExtendedHealthCheck.hc where the content is returned.

I've searched for something similar online but can't quite figure it out. Any ideas?

Thanks!

-Samson