@Lee ************
you apply for the extension at the immigration office that serves the area where you live. It has nothing to do with where the mother and child are listed in the thai house book listing it has to do with where you all live.
I'd say take the mother, child, the relevant bank statements, documentation photos copies of your lease and your passport to the immigration office at Chaengwattana and see if you can get an extension.
You will probably be required to file a TM.30 (which you can do by yourself without any documentation from your land lord other than your lease) and you'll probably get an 800baht fine because you needed to file a TM.30 within 24 hours of moving BUT then you should be able to get the extension.