Entering the US with dual citizenship but US passport is long expired?

7 weeks should be enough time for a new passport. I would suggest you contact your local US consulate immediately and get the process started, letting them know of your time constraint. There are likely to be ways to expedite the process if necessary, but I don't think it will be necessary.

As for the agency, I would just explain to them that you have applied for a new passport and you will give them its number as soon as you get it. There should not be any real need to have this information until you actually check in for your flight.

I just checked the processing times for a US passport application at a few embassies:

• London, approximately four weeks (faster service at consulates general)
• Warsaw, five to ten business days
• Prague, eight to ten business days
• Rome, approximately three weeks

I just checked the appointment offerings in London as an example, and they have 38 spots open tomorrow. They have at least one spot on every weekday this month except for the 9th, 16th, 26th, and 30th.

Wherever you are, seven weeks is more than enough time.

While indeed there's enough time for a passport, since we are a QA site let's review whether flying with this combo is feasible. The problem is not the border because citizens are let in one way or another and an expired passport is enough to prove citizenship, the problem is at check in. The question is whether airline would let them check in with a valid passport to prove who they are and an expired passport proving citizenship. Airlines are required to fill APIS data for everyone departing to the USA (and AFAIK APIS shows them whether a passport has ESTA or not) and the eAPIS portal does support entering two documents:

There are some rare instances where a traveler may choose to have two travel documents submitted (most likely an alien registration card number and a passport) on his/her behalf. When a traveler has an alien registration card number, it must be submitted as the primary travel document.

source and the same PDF details validation elements and it would seem it's possible to enter an expired document, only the well formedness of the data is validated. I do not know whether the version used by airlines support the same. Contacting the CBP is in order, I will do so and report back.

Here's a worksheet from that PDF showing two documents:

I didn't think this would work, but I successfully bought a plane ticket on a Canadian Passport, and went through immigration on a EU passport (both current.) There were multiple windows open for EU passports, and a single for non-EU citizens. I would have missed my flight if I waited at the Other Passports window, and decided to chance it at the EU terminals. Went right through, no extra questions asked.

This was in 2018 or later at a human-staffed window, but sadly I can't remember at which EU airport.