You dissolve SrCl2, which is not volatile, in a fuel that is somewhat volatile, and send the solution in a wick to evaporate them at the flame's heat so their vapour burns.
I'd would find logic (but this has little value versus experimental evidence!) that the wick evaporates only the fuel but very little of the salt, which accumulates on the wick.
Salts can color a flame made by burning powders. These don't rely much on evaporation, and they provide intense heat immediately at the powder.
But in a fuel burnt gently with air, I would not use such a salt, because it's not volatile enough.
Search for strontium compounds that are volatile if some exist. Could be small organic salts like diformate or diethanoate, or Sr(OCH3)2 -if they exist and can be purchased or synthesized. The chemists here may hopefully bring more ideas. Then, the dissolving fuel (if any!) would better be about as hard to evaporate as the strontium-bearing molecule, to avoid segregation at the wick. This was my attempt with strontium distearate, but the distearate may not be volatile enough even if used alone, hence the smaller molecule.
With iodide, I fear the heat destroys the compound before evaporation. You'd get iodine or its compound in the air and a strontium compound on the wick. This may even happen a bit with the chloride.
The other path would be to introduce the metal, as a salt or metallic, directly in the flame at a location hot enough to evaporate some. Like it's done with a crystal of table salt. But does this fit your needs?
If you succeed, you'll get a strontium compound (the oxide, a salt...) in the air. Depending on the use, please check the toxicity.