Just to be clear, what you have been offered on the phone is not a BAH but Avios Hotels. You can find the link on the BAEC Rewards App and/or BA Shopping Portal (rewards.ba.com - you need to be logged in).

I recently booked directly (not on the phone) and the Avios posted quite quickly after the stay. I also found that although the price was the same as shown on Expedia for a basic room, I actually got the confirmation for a room three levels up against the Expedia’s price points.