That's gonna be a tough one to call, as it sounds on the surface that it comes down to misunderstanding and miscommunication. It emphatically underscores the importance of making sure that throughout every step of an project engagement, all parties are on the same page with a clear understanding of the terms and the "rules of the game".
For example, you mention an assumption on your part regarding the billing practices in the client's industry. Unfortunately, assumptions can and will bite. Better: pow-wow with the client and get a solid understanding of their billing arrangements. The more you know about the client, the more valuable you'll be to the client.
With respect to your present situation, it'll most likely come down to a judgement call on your part as to how best to proceed. Re-examine all the facts from the beginning, all communications with the client, and so on. If you genuinely believe your firm had given the client all the info it needed to fully understand that there would be substantial invoices coming from you at a later date, along with at least a rough idea of what the magnitude of those invoices would be, then you might consider legal action.
That determination will, of course, have to include an assessment on your part of such factors as the materiality of the disputed invoices, the importance of retaining this client, public image, and so on. In addition, you could probably obtain an attorney's opinion on just how strong or weak your case might be, before you proceed.
Going the other way, your assessment of these factors might lead to the conclusion that the wisest course of action is to take the loss and chalk it up as a valuable lesson learned, namely, making absolutely sure the client always has a realistic expectation of what lies ahead, at each step of a project.
There's also the possibility of some middle-ground alternative. Suppose you decide that some fault lies on both sides of the table; e.g., you have documentation showing that the client was clearly on notice that invoices from you were forthcoming, but perhaps your communications weren't particularly strong on the expected dollar amounts, such that the client might reasonably have been left with inaccurate expectations regarding their magnitude.
In such case, negotiation might be in order. Maybe you could reach an agreement that the client pays some % of the disputed invoices, you agree to waive the balance, and you commit to the client that henceforth your communications with the client will leave no room for misunderstanding.
But again, only your firm can determine the best road to take, after re-assessing all the pertinent facts. Best of luck with it!
