Since there is no access to the Java proxy in order to be able to generate a client certificate, I do not see how this will be possible.
How secure is the connection if you use firewall rules to allow only Google servers for your jdbc port:
nslookup -q=TXT _netblocks.google.com 8.8.8.8
On your mysql server, configure the user host component to also allow only these ip ranges, and just in case, set a secure certificate size password.
A person in medium attacks will be the only real security problem you are experiencing, not sure these days how likely a MIM attack is ....