As an extra measure, I make an estimate of how many bottles I'm going to fill, half fill most of the bottles in a first run and then go back and top up the bottles in a second run in the same order. I only use 750mL (25oz?) bottles and would imagine this would be more onerous if using 350mL bottles.
My method has been to pour boiled sugar water on beer in the bottling bucket, do gentle hour glass swirls with the bottling wand to mix and leave it for half an hour or so to mix properly (while I carry on with other prep).
I can't pour the sugar mix into the bottling bucket before racking the beer into it because I need to know how much beer there is to prime in order to use the right amount of sugar - chicken & egg problem
In some cases I do know exactly how much beer to prime so do the sugar in first option then.
I do wonder though how much time would be enough for beer and sugar solution to mix sufficiently through gentle stirring as you would want to minimise the amount of time the beer is exposed to oxygen.