1 Apache RP works, 2 Apache RP does not

To persist on your own cookie (not the BIGIP cookie) then you can use a universal persistence profile. This example is for JSESSIONID, but you can change it to your own cookie name:

https://support.f5.com/kb/en-us/solutions/public/7000/300/sol7392.html

Cookie hash persistence would also work:

https://support.f5.com/kb/en-us/solutions/public/7000/000/sol7019.html

And also just normal Cookie persistence will work, but it will result in an additional cookie being used.

Whichever method you choose, you should set the persistence on the second F5:

Internet->F5->Apache RP x 2->__*F5*__->App Server Web Tier

The other devices should not need persistence, only they need to proxy cookies.