Interesting the modest legal advice given was under your national laws and none under Qatari Sharia based law.
As that is the guiding precept in Qatar, all parties concerned are in Qatar and the dispute is in Qatar, that should also be under consideration.
First consideration: What is the age of the child?
Second consideration: What is the sex of the child?

Then consider Sharia law. I suspect strongly that the age and sex will fall towards the mother in custody. But that is just a well tuned instinct to such matters...

But on a personal note: WHATINHELL happened? You two apparently loved one another enough to create a life between you. That is a responsability, not a parcel of property!
My wife and I have gone hungry enough that I fell very ill during our first years of marraige. I made sure she ate more than I, as she was pregnant with our first child.
Things got a fair amount better over the years, but far from comfortable, in spite of the BS everyone hears about US people. We BARELY make rent here and we are sharing the villa with 3 couples and one westerner (I get half, he gets half of the bills. HE collects whatever from the Filipinos here). We raised two children before she went barren and they're grown. One still requires occasional support (more occasions than not). After nearly 27 years, we are still together.
So, a simple question, but first: Love and hate are the same at the end of the day. Hate is simply a response to rejection of love. Not an absence!
Was and is there now any love alive between you? Remember your RESPONSABILITY to your child! Either way!
If there is, I suggest that you both stop BSing around and ACTUALLY TALK to one another. NOT emotionally, but honestly. Not as loyal wife and supportive husband, but as equal partners. In a household, you are. You should be in charge of the house, he handles the other matters. It worked for us in the US all these years. And I had her handle the money for the house too, as I am horrible at mathmatics.
My wife can say nearly ANYTHING TO ME without offense. If it's offensive, I have to give extra consideration as to WHY she said it. The same goes with me.
Is it hard? HELL YES! EVERYTHING worth ANYTHING is hard to attain and keep. The battle to keep it going is worth the effort!
So, is it worth the effort to try to revive what you once had? Yes, it'll be tough, it'll be double-tough. Is a fine, strong family worth it?
My parents thought so when they shared a can of condensed soup between them for their daily meal. My wife and I thought so when we shared three frankfurters between us.
But what we had was WORTH protecting for ourselves and our children. The question I put before you, first disregard what was done AFTER splitting up, that was a reaction to the split.What you HAD, is it worth recovering? Trust me, from personal experience with helping a number of friends, you would be amazed...
But, as in all things human, it's your and his free choice. Just remember you will be judged by your CHOICES, not as much the results.