Not sure if the OP is coming back, but never mind.
If you get a passport in their current name, plus a letter from SS giving permission to travel, you may be OK. But obtaining this may be difficult, and may well require the BPs permission too.
Otherwise you need to wait until the adoption order is through. Normally you can't apply to court until at least 13 weeks after placement. With an older child the SS may not support an application until some time after that (we had to wait 9 months with similar age DD). BPs may decide to contest adoption, unless you have very specific reasons where you can be sure they will not. You may nopt then get a court date for 2-3 months.
Once you have been to court, you will need to wait to obtain the long birth certificate. iirc that takes around 4 weeks or so. Once you have the long certificate you can apply for a UK passport, this again is not going to take less than 2 weeks, probably longer, unless you can go and queue for one in person.
If you have the AO and a British passport in your surname she should have no problem going to US as she should be treated like a birth child.
I think you need to talk to your SW about the likely timescales and go from there.
Personally I think you are being very optimistic if you think the AO will be made in time for you to get a UK passport for a trip this year. But you may have information that you have not shared that means you are not being so.
I think organising to travel with a LA child to the US may be a bureaucratic nightmare from both the UK and the US side.
Normally I would also say that expecting a 7 year old newly-placed child to go on a transatlantic trip would be highly risky as to whether said child would cope. Ours definitely would not have coped. But in your case, given you already know the child, you are maybe in the best place to judge. But you should not judge all the kind helpful adopters on here who gave you best advice on the limited information you initially provided.