The ClF₃ is produced biologically, just as fruits produce acid or capsaicin. This alternate world just happens to have bacteria — and later, plants — that hit on the trick of using fluorine to produce toxins and novel organic compounds. This has some precedent in the real world. Just as the early civilized people produce gelatin, lye, and other chemicals through mixing, cooking, and processing natural feedstocks, they learned how to produce ClF₃ using especially useful biological precursors.
So, how can they make containers that hold it (to be thrown at enemies as a weapon) and how can the wielders of this weapon protect themselves?
Note in particular that chlorine trifluoride will ignite most organic and inorganic materials. You might suppose that the availability of fluorine-bearing molecules in the local biology will offer a solution, but the obvious Teflon is also ignited by the stuff.
Note: see this meta post
I'm not too picky about what “Bronze Age” means exactly. It covers a lot of time and different cultures in the real world. Only the general ideas: the people are well into agriculture, have metalworking but not the temperature and technique for Iron yet, wide-range trade, and inventions of “devices” like pottery wheels.
Clearly this alternate-reality culture will develop differently due to the fluorine biological resources. Taming ClF₃ (for warfare) might enable the iron age immediately thereafter. So, don’t worry about the exact age and culture — it will be made up anyway.
Let me re-iterate: the plants/bacteria/whatever do not secrete ClF₃ directly. I realize that it doesn’t offer obvious evolutionary value. Rather, the plants produce “usable flourine compounds”, so that primitive people can produce the stuff by mixing and cooking (and waving of hands).