At the risk of sounding silly - Instead of focusing on burning the solids, boil the water. Water boils at 100C, at which point the water vapor should separate and leave all the solids behind. Then capture the vapors and condense it back down into clean water. Now, if you later want to incinerate the leftover solids, sure, go for it, fire’s always cool in my book.

I’ll add, simply boiling water is energy intensive. What you are proposing probably won’t work at any scale.