I’m not a Chemist (and too lazy to do some in dept research about that ), but a good idea and it sounds that you made more powerful gas.
I guess due to energy conservation, you may only equal the amount of energy of methanol in the gas. With some unavoidable losses it is probably less than adding it after the gasifier to burn in the engine directly.
On the other hand, the extra oxygen and its ability to burn in the gasifier inlet, thus bringing extra heat in the reaction, is definitely beneficial for the gasification.
Rindert, adding exhaust gas in the gasifier does not result in richer gas but in a better heat economy (you burn less char for the same result), as you recycle some of the CO2 instead of burning fresh char and wasting heat. But for richer gas, you have to create hydrogen. But in the exhaust gases from dry charcoal gas (no water added) are just traces of water vapor, so this is not the way to create more powerful gas than with just plan air.
This is just theory, in practice some other effects may give slightly different results, but in general you have to produce more hydrogen for more power.