Would the US government allow us to travel and for my fiance to give birth in the US?

First of all, there is no law that directly addresses this situation. For years, people have come from the Philippines, Russia and elsewhere to give birth here. In the past few months, however, I have heard of several instances at the Houston airport where the immigration officer refused admission to pregnant women with a B-2 visa. So, I'd definitely suggest avoiding Houston. Whether this is happening elsewhere , I don't know. if your wife is not "showing," then they'd probably let her in, but there's never any guarantee.

You cannot just plan to have your child be born in the US. Most likely, if the mother is in the advanced stages of pregnancy, she will not be permitted to travel. And even if she is allowed to board the plane, she may not be permitted to enter the US if the only reason for coming to the US is to give birth.