Isn't it easier/simpler to just ferment it with the added sugar and yeast in the bucket, using a lid and an airlock? That will make it alcoholic.
When that's done, which should only take a few days, add a
measured amount of sugar* to the bucket, mix carefully, and bottle. In 2-3 weeks all bottles should be perfectly carbonated. That way you won't have to depressurize each bottle several times and risk them exploding due to overpressure.
* Here's a carbonation calculator for bottling:
Calculate precisely how much priming sugar to add at bottling time (bottle priming) for your home brewed beer. Works with a variety of priming sugars.
www.brewersfriend.com