US Citizen with dual nationality entering US on ESTA, what's the worst that can happen?

So far, the existing answers offer frame challenges, but your actual question has only been addressed in comments. So let’s compile those comments into an answer (because comments are ephemeral).

What kind of bad stuff can happen when lying on an ESTA application?

• Your ESTA gets denied, you get denied boarding by CBP, you get a long talk with CBP on arrival, you have to wait for hours on arrival for them to check your documents. (@jcaron)
• As to being denied check-in with an approved ESTA, it’s possible that the CBP system that evaluates the APIS record sent by the airline will notice your US citizenship and send a "do not board" response because of 8 USC 1185(b). I don’t think they do this, but who knows if they’ll start at some point. (@phoog)
• The worst case scenario seems to be a fine of up to $250,000 and/or imprisonment up to 5 years. This is an exceedingly unlikely punishment even in the exceedingly unlikely event of a conviction under 18 USC 1001, but it is theoretically possible, so qualifies as a worst-case scenario. (@phoog)
• The main problem with a conviction under 18 USC 1001 is not the penalty or imprisonment (which may be unlikely), but a felony conviction, which is an indefinite stain which can cause a whole lot of troubles to the person. Ask Michael Flynn. (@littleadv)

The best solution would be to get an emergency passport from the nearest US consulate.

If you cannot for whatever reason, you can travel through Canada or Mexico, and use the Global Entry card to enter the US through the land crossing. Given the documents that you said you currently have, this is the only way the CBP would let you in without an emergency foil from the consulate.

For reference, see the CBP guidance here. If the CBP inspection raises a question as to why no US passport is linked to your Global Entry card despite your naturalization, you may explain the situation to them. While it is very unlikely, they may still send you back to the nearest consulate for an emergency document. I doubt that would happen, since the Global Entry card is an acceptable document to enter the US by land, and your Global Entry card continues to be valid. You’ll update it with your US passport once you get it at any Global Entry service center.

Update: looks like OP reported that getting an emergency passport was super quick. Updating my answer accordingly.

What I would do is:

1. See how fast you can get an emergency passport. If it’s quick enough, go get it. Otherwise…
2. Apply for ESTA, specifying you’re a US citizen. Some people on this site have reported it works.
3. If the ESTA is approved, you’re good to go.
4. If the ESTA is rejected, buy a ticket to Canada (Vancouver or Toronto are a good bet) or Mexico (Tijuana), then take the bus, train or drive a rental car to the US. If you go to Tijuana you could also just walk across the border very easily thanks to the Cross Border bridge.
5. Since October 1st 2022 VWP-eligible passports are required to apply for ESTA in advance, so taking public transit might be a little more tricky, as some bus operators verify your documents in advance, though it’s not clear if they now also check that you have a valid ESTA. So if you don’t want to drive a rental car, your best bet would be to use an operator that doesn’t check IDs, take the Amtrak from Vancouver (where there’s CBP officers at the station) or go to Tijuana as mentioned above.

Once you’re at the US border the CBP officer will pretty much have no choice but to let you in. As an absolute worst case scenario you’ll have to wait a bit in secondary inspection while they verify your naturalization details.

Instead of trying stunts like lying on your ESTA which have potential consequences, contact the nearest US embassy and seek advice. You are not the first US citizen to be abroad and need to get back quickly without a passport.

You can also try applying for ESTA and disclosing your US citizenship, which protects you from accusations of lying to the government (which is potentially a felony). If your ESTA is granted, then you can fly to the US with your French passport and present yourself at the US citizens’ line when you arrive. Show your French passport and your naturalization certificate and explain why you didn’t get an emergency passport to return to the US.

Another option is to go to the US embassy or consulate in France and get an emergency passport, but this might cause your pending passport application to be abandoned.