As far as I know, dissolved CO2 in a liquid is the same whether it has sugar or not, BUT it could be that the flavor effects change with sugar content, or the sugar reduces the ability of the liquid to absorb CO2. I don't know. But comparing the
2 charts here, they are not very different, sugared versus sugar-free. Worst case I can see is less than 0.25 volume CO2.
But it is a simple gas law thing: You are dissolving CO2 in water. How much you can dissolve changes with temperature. How much IS dissolved changes with pressure. What it dissolves as, exactly, changes with pH. I can imagine what is in the water has an effect, but it must be small.
For carbonation, read the part labeled "Carbonated Beverages"
here.
For one such formula, look
here. Read it all the way so you understand the formula given almost at the bottom. Rearrange it as needed to solve for Volumes CO2 at a given pressure (the formula shows the pressure for a given CO2 volume). There are others out there too.
All that being said, if you use a CO2 tank to carbonate your beer in a keg, do the same with water. Maybe add just a touch of a flavoring if desired, a little goes a long way. And after a few days, try it: Not fizzy enough? Increase the pressure. Too fizzy? Reduce pressure. In other words, the tables are a good guide, but your taste should be the final arbiter.