OK here's the thing. You basically have two choices when carbonating your soda: natural carbonation and forced carbonation. Since natural carbonation appears to be no good for your purposes, you're stuck needing equipment to force carbonate.
I totally understand that buying a CO2 tank just to make halal root beer is kind of a big investment.
HOWEVER you might be able to force carbonate using a similar setup, but with two bottles, each with their own carbonator cap. One with the soda , and another in which you mix baking soda and vinegar to generate the CO2. You'd have to be careful about quantities in order to control the amount of CO2 you generate.
So basically make two carbonator caps using this method:
And then connect the bottles together with a hose and shake up the bottle with the vinegar and baking soda in it.