[RedDevil]

I'd go for a very small engine, like a model aircraft engine. Then pumping loss is less of an issue. And those are designed to run on rather explosive fuel anyway.
Maybe use alcohol or a a gasohol blend as a fuel, mixed with water to keep the temperature down (instead of injecting it). Water is readily available (the X-prize concern) and should not count as fuel(*).

Then again, instead of pure oxygen you could use a hydrogen peroxide mixture just like the Messerschmitt Me 163. That though lost it wheels as soon as it took off, a feature you'd probably want to avoid for your daily driver.

(*) The X-prize requirement does not seem particularly stern. They could simply claim that anything you'd have to replenish should be counted as fuel.
Once nuclear fusion plants get scaled down from star to car engine size, plain old water would definitely be seen as fuel.