What can I do against a competitor copying my software?

Copyright infringement is actually one of the easier cases to prove because as registered owner, you have providence (proof of how you created it) and comparison whether identical binaries or substantially similar with superficial changes can be trivial to demonstrate (though compilations or mixed use becomes increasingly complicated). What becomes more harder is separating the expression from idea where they've reimplemented the functionality via clean room techniques (which is legitimate). So whilst it is easy to claim, there are certain defenses (eg fair dealings in Australia like backups) or compatibility/interoperability. There's also a recent Australia law (reminder to self to check progress) that allows SMEs to escape some of the more burdensome contractual adhesion clauses by dominant vendors.

As for evidence, basic property theory, proof of legal ownership, evidence of infringe (eg buying competitor product and showing direct relationship to copyrighted material), perhaps expert opinion on the degree of infringement, evidence that you've asserted your claims against the right legal entity (infringement could be done "unknowingly" by subcontractor) ... as always consult a lawyer if you want to go down the legal route rather than settlement.