You'd need a "path of least resistance" setup for the exhuast, so that it could not come back through the vent under low pressure situations.
As was said, yes, it could be done. Special care would need to be taken, though.
When I first read this, I read it as (duct radiator through exhaust) meaning: (run coolant through a radiator fitted to the exhaust pipe to cool the engine). I was all prepared to tell you why it just wouldn't work.
The added benefit of running your heated radiator airflow to the exhaust is that:
At low radiator velocity, high exhaust velocity, the exhaust will act like a siphon flow, pulling more air through the radiator via negative pressure at the back of the radiator's surface.
At high vehicle speed/low engine speed - the excess flow from the radiator, ducted smoothly into the exhaust pipe, would work to increase scavenging due to a siphon flow effect creating negative pressure on the back of the valves, increasing the pressure differential across the face of the valve as it opens.
At a high enough speed with low enough engine speed (not even extremes, here), one could experience lowered pumping losses due to the piston no longer having to apply nominal pressure to the exhaust gasses to force them out.