It makes no sense. FTVs are available all bookings whether made directly or with a TA. For TA ones they FTV is done through your TA though. There is no exclusion from BWC/FTV for your booking which I can see here.

Everything would normally be on an FTV, so cash, avios, and cash for the UuA. As you will have done your UuA with BA it may be worth calling them to see what they say, but they may well probably tell you that you need to speak to Amex.

I think it very clear Amex have lied and messed this up for you completely.