As far as I know an atomizer, mister, and pump spray are the same thing. I use the Muji 12 ml. It's the smallest I've ever seen, but for cocktail purposes this will last you for months, and the pump works very well. I had one from the Container Store that often jammed for some reason.
I think spraying is the best way to dispense bitters. In particular, spraying some bitters on top of a drink really brings out the nose. This also has the interesting effect of causing the drink to change drastically over time as the initial sip is very bitters forward and as it settles this goes way.
You can also do an absinthe rinse easily, though I always prefer to spray on top of the drink.
I keep spray bottles of Dr. Adam Elmegirab's Boker's Bitters, Angostura Orange, Bittermens 'Elemakule Tiki Bitters, Laphroaig, and absinthe.
Do spray bottles completely obviate the need for dashers, such as the ones that come with the bitters bottle or glass dashers? It's hard to measure, but I think one pump is roughly equal to a dash.
I just found that the Muji website has a warning: "Anything with a high percentage of alcohol should not be placed in these containers. No foods, drinks, oils, nail polish remover, perfume, cologne, toner ink, vinegar, paint should be put in these containers." I'm not sure if they're just trying to cover their butt legally, but this article suggests that PET is fairly safe. The amount of liquid dispensed through a spray bottle is trivial compared to a wine bottle. You could get a glass spray bottle if you're really worried.