I'm not sure if you can apply for ILR now, or if you will need an extension on your FLR (the experts will know this) but whatever you do, you must do it before the first of May.
I'm hardly an expert, but just from doing the math...she can't apply now because she doesn't qualify yet (ie, she's not within 28 days of the 2 year anniversary of entering the country.) She will qualify on 28 April, a week from tomorrow, which then gives her just a couple of days to sort out her visa.
Option 1- send in ILR by post and postpone/cancel the trip.
Option 2- try to get an in person appointment between 28 April-29 April. If successful in getting an appointment and a visa approval, travel on the trip as planned. As I said, this is probably going to be impossible (the PEO's are almost always booked a few weeks out) unless they have a late cancellation, but absolutely worth a call just to make sure.
Option 3- go on the trip and apply for another spouse visa in the US before returning to the UK. It is an option, but in my opinion doesn't make much sense when ILR is an option now, especially considering the rule changes that lie ahead.