How to determine if an unknown is potassium chloride or potassium bromate?
Noting the solubilities of potassium chloride and potassium bromate gives a nice point in the identification but apart from it, there are certain chemical test specific for certain anion that gives pinpoint information about the salt.

First is the chromyl chloride test for chloride ions. Perform the test and you will identify potassium chloride (potassium bromate will give -ve result to the test). Second is conc. $\ce{H2SO4}$ test. Potassium chloride will gives vapors of hydrogen chloride. Potassium bromate will give bromine gas.

$$\ce{KCl + H2SO4 -&gt; HCl ^ + KHSO4}$$
$$\ce{4KBrO3 + 2H2SO4 -&gt; 2Br2 ^ + 2O2 ^ + 4K+ + 2SO4^{2-} + 2H2O}$$

Third test is the heat test. Potassium chloride is stable to heat while potassium bromate is prone to decomposition $\ce{\approx 370°C}$.

$$\ce{2KBrO3 -&gt;[\Delta] 2KBr + 3O2 ^}$$



About the silver nitrate test pointed by @Dallon Penney, precipitation will occur in both cases albeit depending on the concentration of respective ions, it seems that the precipitate forming has different properties. Silver chloride exist as a white, curdy precipitate, insoluble in water and in dilute nitric acid while silver bromate exist as a white crystalline precipitate soluble in hot water and sparingly soluble in nitric acid. Both precipitates however are soluble in ammonia solution.