Arabella:
I don't mean to come down hard on you. I hope you get my point; my advice is avoid this local marriage idea unless you do it according to your own faith and it is the real thing once and for all and not a trial run. And I'm not speaking from a religious viewpoint as I am neither qualified nor committed.
There are many unmarried couples here. As Angelwings pointed out it is illegal to co-habit in Doha and the minimum punishment is lashes and deportation. That's according to the same Sharia law by which you think you might want to get married. Also for a Sharia ie Islamic marriage at least one partner must be a Muslim.
Co-habitation exists and I'm not aware of people being prosecuted for it. I know 2 couples who co-habit and they seem to be doing fine. Unmarried couples, not necessarily co-habiting, go out shopping, to the movies, restaurants, dancing, to the beach, etc. and no one is stopped and questioned as to who they are out with. It's not Saudi Arabia here.
You decide.
I think the biggest hassle here after the initial and obligatory red-tape of residence permit formalities will be traffic and some of the so-called drivers on the road. And watch out for the humidity.
I hope you guys have a good start in Doha. I wish you well.