So I guess your little one will not get a US Passport right away from the US Embassy in ADD? In which case, the US Embassy must have experience with this situation may be able to tell you which possible transit country missions in ADD will issue your child a transit visa with the least delay & hassle. LHR & DXB have a frequency advantage but if you are a DL emp you have many options: the Schengen, Israel, Egypt, Ghana, Senegal & South Africa. Combined, a Schengen visa would give you most options.

As an alternative you could buy a refundable confirmed tickets from the transit country to US to meet the immigration requirement and refund it when you all get home. Down side is you'd tie up thousands of dollars on your credit card until you get the refund.

I'm trying to get to ADD later this month & it seems the IAD-FCO-ADD direct is very heavily booked all month.