How do I configure a simple TCP proxy that can rewrite SQL from a client to a MySQL server?

Although this is technically something that could be done with an iRule, using binary scan, it would be a very complex task. You would effectively need to reimplement the MySQL Client/Server protocol in an iRule, both to decode the incoming packet, and to create a new one.

It may be that for your particular needs, you could reduce the requirements to only meet what your application uses, but doing that could also limit your options in the future.

Perhaps this is something that could be implemented by modifying the client application, or using mysql triggers on the database instead ?

In future versions of BigIP, iRulesLX should make this sort of task easier, as you would be able to pull in a mysql client library, but currently this is an Early Access feature only.