Perhaps not. This
POST from 2017 includes an example of a Premium Select itinerary with the domestic portion in Economy because the required fare bucket was not available in First at the time. This would explain why you see domestic First on some flights, but not others, on the same day. This isn't Amex Travel, it's supply/demand.
In that example it sounds as if Delta was open to the possibility of moving the passenger into First if the bucket opened up, but they may not be willing to touch an agency booking.
Yes that might be the case. However before I searched on Amex's website and made the booking I check on Delta's and it was the same price (with first class) as the later flights (which if i booked with Amex will be first as well.) And that on my current reservation on Delta's website only 1 of the first class seat is taken (it is in January). Delta is also willing to sell me an upgrade to first class for 211.5 USD.
I messaged Delta about this, they told me there is nothing they can do...