Welcome to the Community @krishnaramani
That is as amateurish a mistake as I have seen about an agent. Whether rights come into play or not, you stated what you wanted and accepted a quote believing they met your specification and FC being one of if not the largest agency in Australia would reasonably be expected to know their business, including issues with date formats, especially with the US as well as knowing about issues with 12 versus 24 hour time norms.
The US is essentially an odd one out normally using 12 hour time and mmddyyyy formats, and even if done by a trainee agent, they should be aware and their work QA’d prior to being sent to a customer.
In our experience agents invoices clearly show the included products even if not as official tickets. Was your invoice a single line item or did it show line items with dates? If it showed dates were they all in mmddyyyy format or were the rest of the arrangements in ddmmyyyy format?
My opinion is FC did not provide the expected care and skill a reasonable person would expect from a travel agent.
See section 4 of this ACCC document.
You could ask them to honour their quote because it was their error. Alternatively you could ask for a full refund. Asking for a full refund would probably be counter-productive even if they agreed (which would be unlikely) because the air fares and hotels may have all risen in price, as well as the Disney tickets, if you were to rebook.
In honesty and as something to consider if they decline to cover their error, would you have accepted the ‘correct higher cost quote’ or would you have declined the entire package or part of it?
In your place I would have a chat with the office manager and if getting push back I would formalise it with a letter using the form of a ‘Letter of Complaint’ asking for them to cover the difference, with reasons why they should. You can find substantial information on a Letter of Complaint searching the Community, on the main Choice web site, and on the ACCC web site.
If not amicably resolved and you wish to go further, your state Fair Trading/Consumer Affairs office would be your point of lodging a formal complaint. Note that the Australian Travel Agent Association of which FC is a member is a lobby/voice for the travel industry, not for consumers.
Please let us know how you go with a resolution.