Hi, we have a similar story. Also pregnant with baby due soon now. Didnt want to give birth in my home country (Brazil-- I'm the beneficiary). Every lawyer my husband has spoken to said the chances of an "expedite" or something else due to pregnancy were slim. I also tried tourist visa (very early on in pregnancy), got denied. Thought of trying again for the tourist visa, but then with the pregnancy more advanced, I learned that the US is no longer accepting "visa babies" or whatever is called, aka people who go to the US only to have the baby and leave. We literally looked at options everywhere in the world... Canada, Europe, other places in south america, etc.
Now for some unsolicited advice, coming from someone with a very similar background as you... If your wife is from Egypt, she probably knows the best places/hospitals/cities with the best hygiene rates etc. Same as me, I know which places in Brazil have the best hospitals (actually anywhere really, as long as you can pay, which is not a lot either), and thats what I did. I WISH we had spent less time trying all sort of ways to go to the US to have this baby (extremely unlikely unfortunately), and instead focused on finding the BEST city, hospital, team to deliver the baby. It would've saved us a lot of stress and headspace. Just my $0.2