Ideally, you want the propellant to go straight out the back as fast as possible. Any sideways motion, i.e. from heat, is wasted kinetic energy.
That's one way to think about what a conventional rocket engine's nozzle does: By expanding the exhaust, it cools it and converts that (sideways) thermal energy to desirable velocity toward the back.
So if you burn your propellant before running it through the ion engine, you'll lose performance unless you find a place to put an expansion nozzle before the ion engine. And since it's an expansion nozzle, you'll need a Really Big Bore on the ion acceleration channel: That adds mass.
You'd be better off using the burning fuel in a conventional engine, dropping it away, and then starting an ion engine. Each could then be optimized.