Well, the carbonation tabs do work but not quite in the way you describe. The carb tabs are a mixture of extract and dextrose, so you add them at bottling just like you would add priming sugar. You must have the yeast to bottle carbonate it, since that is what causes the carbonation. There is a mini-fermentation in the bottle, and that does leave yeast sediment. So, if you filter, and try to bottle carbonate, it won't work. Your beer will be flat. If you add yeast at bottling so you can bottle carbonate, then you'll have sediment in the bottle.
One thing you can do to reduce the sediment is to use a clearing tank after primary fermentation is over. That does help with the amount of sediment. I have very little in my bottles. The other thing you can do is use a very flocculant yeast- it will compact well in the bottom of the bottle and so it "sticks" in the bottle than others.