Indeed, the default management certificate does not include a SAN for the management IP. In your place, I would replace the management certificate with one for which you can control the trust chain. Tipically this would be an Active Directory CA chain, or similar PKI.
If you don't have a proper PKI available you can try a "light" version, like TinyCA (which is a simple graphical interface for openssl). Or pure openssl commands if you're into that!
Either way, you must import the CA trust chain into java trust store. If you want to use the IP address to access your F5 system, don't forget to add something like "IP:10.10.10.10" to the SAN field of the certificate.