Yeah, as @
Andrew7447 said, it's such a small amount to boil. Once dissolved and added to your bottling bucket, it has cooled down to room temps (or close) already.
Sugar dissolves much faster in warm or hot water, so there's that time advantage, instead of stirring it for 5 minutes.
In a similar light, many extract brewers doing partial boils top up their boiled wort by adding a fair amount of cold tap water to it, like 50/50, sometimes more. Then pitch yeast. That's much more than a cup worth, meaning more microflora.
At least you have alcohol in there. If that's water you'd drink, I think it would be fine.
Now, I would be more concerned about the bottled water
closure itself, the lid, threads, neck, etc. So sanitize those well. Spray or mop Starsan around that area before pouring the water out.