From this article:
Taking Listerine pocket packs® (that have a hechsher) is not permitted
on a fast day since one swallows the liquid which melts in the mouth.
(All year-round a shehakol should be recited on them. Many people are
un-aware of this and they should be told).107
While the purpose of the Listerine is not refuah, the ingestion method is similar to a throat lozenge. Like Listerine, the throat lozenge has flavor, it melts in your mouth, and you are swallowing the melted sugar liquid. Note that in the citation, it states the necessity of saying a bracha on Listerine, similar to what you indicated with the throat lozenge. So, it seems that they are definitely considered "food".
If a doctor specifically told you that you MUST take the lozenge (I'd be surprised, as you could probably accomplish relieving the sore throat by swallowing a pill or some other permitted fast day method), then there MIGHT be a heter for doing this.