I'm a subcontractor that had to file a stop notice on BBII. The public entity already confirmed for me they are withholding the funds from BBII, but also tell me it's up to them to pay me not the public entity. It's been almost 2 months now since they received my stop notice and I still have not been paid.
To "perfect" a stop notice, you have to actually sue to enforce it. You'll probably want a construction attorney to help you draft the lawsuit and enforce it. You probably also have breach of contract claims against your GC that you would include in your stop notice.