I'm with the oldglassman about it not being a bitters bottle, Your bitters bottle is really fabulous and would be at home in any good vintage bar or museum.
It really looks like a nozzle though, throwing a thought on the fire, apothecarys shops used to be a bit more cafe-ish with a culture of come and consume, I can imagine this nozzle on one or more of the many ingredient vessels, a drop of aqua mirabilis from a small fancy container with a ruby like nozzle would certainly put up the price and make the consumer think they were getting a special medicine.
It certainly would restrict flow so am thinking dropper nozzle.
Loving this tread. I really don't know what it is but it's the finding out that's the fun