Here is somewhat a legal question.
Since there was a signed contract with terms of payment for services rendered, who is in breech of contract? Them for not paying with a notice from you stating services will be stopped until full up to date payment is received or you for stopping services?
That will be their argument that you stopped services.
They are wrong for not paying you or paying beyond the terms set, agreed and signed. Therefore breeched the signed contract.
You are past legal threats, place the lien and move on and never do business with them again.