Efficiency of Flyboard engine
I'm not directly familiar with the engines Zapata uses on the Flyboard Air, but I'd hope they're using a kerosene derivative like JP-1 or commercial Jet-A.  Alternatives (gasoline, diesel, various alcohols) have no better energy content per volume, and some (alcohols) considerably less.

There is no case (for low altitude atmospheric flight) in which a rocket can beat an air-breathing jet for specific impulse/specific fuel consumption -- because of the need to carry oxidizer, instead of just using the air for that purpose, the amount of fuel/oxidizer available is limited by tankage.  This is why airliners, warplanes, and private business aircraft use jets or turbine driven propeller engines instead of rockets (not to mention the far higher durability of a turbine compared to a rocket engine).