As has been mentioned, she is a US citizen from birth, just undocumented.
CRBA appointments gains you the certificate, passport and SSN all in one appointment. Passport takes around 3-6 weeks to receive from the appointment date and SSN currently is taking 16 weeks.
However, people have managed to get in on a UK passport and ESTA but all it takes is a newbie Border Officer, or an extremely strict one, or one just having a bad day to be turned around as is it is the law and I'd class these people as just being fortunate.
Side note: you're missing out on $1,400 of child tax credit per year while your daughter remains without an SSN.