Why ?
All you are doing is taking away all the advantages of a two stroke, namely simplicity and lack of weight and complexity.
Stratified air injection is being used in two strokes to clean up emission's, as are catalytic converters.
Strato engines generally use an extra butterfly and 'transfer' ports to inject fresh air ahead of the incoming transfer charge, this sweeps the exhaust out of the cylinder rather than the incoming charge, significantly reducing the amount of mix lost out the exhaust and significantly reducing emissions.

