Can I threaten to report another party’s solicitor to the LSC if they won't reply?

Practitioners should take care not to threaten the institution of disciplinary proceedings (see Australian Solicitors Conduct Rules 2012 (Qld) (‘ASCR’) rule 34.1.2). You should be sure that the circumstances are such as to warrant a complaint. You should also appreciate that there may be no action the Legal Services Commission (‘LSC’) can take. The other solicitor may be acting on instructions, or there may be some other reason for this lack of response. Such matters may be confidential or privileged and the other solicitor may not be able or willing to provide details, even to the LSC. 

The solution may be to write directly to the other party about the lack of reply from their solicitor. You may do this provided that you follow a certain specific process outlined in rule 33.1.4. Once again, care should be taken before doing so.