It is an official requirement of CX that the passenger name on the passport must match the ticket. The agent was not wrong to say this. However, in practice, the absence of a middle name on your ticket will cause no problem.

My passport:

Given name: FIRSTNAME SECONDNAME
Surname: LASTNAME

All my tickets are booked as LASTNAME/FIRSTNAME MR and my MPC account reads Mr FIRSTNAME LASTNAME.

In travelling to various continents throughout the world, including the US recently, I have never had any problem and the OP has absolutely nothing to worry about.

Since it is an Avios ticket, even if then name doesn't match the BAEC account, it doesn't matter, because you're not going to earn miles on it anyway.