Forum Discussion
I have seen a lot more crazy requirements, and your requirements are not that uncommon.
If my understand of the requirements are correct, I see 2 options:
1 - Portal Access
https://support.f5.com/kb/en-us/products/big-ip_apm/manuals/product/apm-portal-access-13-0-0/1.html
2 - Web Application Access
I can't find any link, but is easy to explain. This is basically APM + LTM, APM is used to authenticate, and after the request is load balanced by the LTM. So, you create a virtual server, link to the APM policy with the logic you need, and is done. Client must point to virtual server, and APM will process first, and when finished, LTM will process the traffic.
In case you don't know, there is also 2 clients you can use for that, in your case the Edge Portal can be a good option:
https://support.f5.com/csp/article/K12488