What would the chances of Qatar starting up at BFS on the B787 or A330 to go head on head with Emirates & Etihad at DUB?
Simple - NIL.
With Qatar flying to just Heathrow, Manchester and Edinburgh in the whole of the British Isles. the chances of them flying to a regional airport such as Belfast are not even remote, they're non-existant. What would be the business case?
The answer to your question was actually in your question - "head on head". The only way you can do that is from the same airport, and no matter anyone's political opinions about north/south, UK/Ireland or anything remotely connected, the simple numeric and realistic facts are that Dublin Airport is a massive player with a large population living within two hours driving time, and more importantly it serves the capital of the Irish Republic.
BFS in itself is not in the same league when it comes to having the pulling power to attract many world airlines. It might rightly be important to the people in Northern Ireland, but while it competes directly with BHD for airlines and passenger numbers it can't build the critical mass necessary to generate sufficient traffic to be attractive to most airlines. In aviation terms it remains, and will remain, a regional airport. Our expectations should be based on that.