According to this article here, QR flyers on eligible fare classes as well as per FF status, should be able to use the Clubrooms lounge at BHX.

In terms of airport lounges, passengers in Business Class (excluding Business Light tickets but including Avios redemptions) can use the Clubrooms lounge. This is, by far, the best lounge in the airport and was where we spent much of our time on Thursday.

Passengers in Economy or on Business Light tickets who hold oneworld Emerald or Sapphire status – which includes British Airways Executive Club Gold and Silver members – can also access the Clubrooms lounge if flying on Qatar Airways.