My understanding of contract law is the original fees are implied if no new explicit agreement is signed and therefore the invoice amount should be $1,000. Is that correct?
No. At the outset, the presumption that the price would be equal to that of filings covered in the engagement letter is inaccurate. The actual terms of the engagement letter might support a different conclusion, though, which is why I asked about them.
The offer in the engagement letter seems to be just a marketing practice, and as such it is neither uncommon nor unlawful. The rationale for that marketing practice is that offering [in this case] the first two tax filings at a lower price is likelier to persuade new customers to try the services the company provides. Thereafter, those customers will be billed what the company would call a normal or ordinary price.
Statutory law does not provide specific constraints to how greater the ordinary price. But the details would help ascertain whether an excessive, unannounced increase amounts to what is known as unfair and misleading practices or otherwise contravenes consumer protection laws. These laws vary by jurisdiction and might or might not defeat the argument that the customer bears the risk of mistake. See Restatement (Second) of Contracts at §154(b).