I will compare with the server side ciphers, but i'm curious to understand how it is working for now. Because the handshake is rejected if I CURL from the F5, but the monitor is UP and clients are able to access the same page.
For monitoring with TCP and making a test with curl, please don't take any offense, but you are comparing apples and pears. A TCP monitor will check if whatever service responds on port 8080, HTTP protocol is not involved at all. Curl will send an actual HTTP request and you will see the HTTP server response.
Can you check, maybe there is an iRule that would disable SSL on the server side under certain circumstances? Something matching this string?