SSLv3, TLS1.0 and cipherstrings...

Without seeing the Wireshark capture, it's hard to say exactly what's going on, but it could be something like this:

http-8443-14, READ: SSLv3 Handshake, length = 87 *** ClientHello, TLSv1

In this case, there is a record of type "Handshake Message" version SSLv3 and length 87 bytes (a typical record size). The contents of the Handshake Message show that this is a "Client Hello" and the client supports up to TLSv1.0.

It's possible that the BIG-IP sees the initial "SSLv3" part of the Handshake Message and rejects it based on the configuration of the SSL profile.

Here's some more info that might help: https://support.f5.com/csp/article/K15292

Also, for troubleshooting purposes, it might help to change the SSL logging level to "debug" so you can see exactly why the BIG-IP sent the handshake failure alert.