Instead of using a complex turbo pump to pressurize the fuel going into the chamber, why not use readily available and relatively cheap diesel injectors. The modern one produces pressures around 300 bar with the use of piezoelectric stacks. They are pulsed at a 50% duty cycle, but you could just put more into the combustion chamber and off set the firing time. You would need to feed the engine a fuel like RP1 and an oxidizer that was liquid. Maybe even feed it diesel....