You wont remove all the sugar in the yougurt. The fermenting process will remove some but there is still lactose in it.
I would be careful and not introduse it to fast,I know it’s temting. One consequens of introdusing it to fast, is that distingusih between die off and allergi/intolerance is hard. If you wait until your done with the die off it makes it a whole lot easyer. I have personally introduced yougert and other products to fast did’t go so well. The amount is inporent to, dont over do it the first time. Some smal sips a day probely would be the best way to start.