You're gonna need a heavy fly wheel to store the energy between pumps. On a rail car the fly wheel is the car itself moving down the track, keeping the wheels turning, and allowing the operators to maintain a smooth rhythm, slowly increasing speed..
If you replace the car with a prop, the first downward push would quickly spin the prop, causing it to cavitate and produce no thrust. Also, because there is little to no rotational resistance on the prop, the handle of the non pusher will fly up and smack him in the mouth, which will likely knock him overboard, and attract sharks, which is usually grounds for disqualification. If he does survive the first pump, his next downward pump would again spin the prop, again resulting in cavitation and reduced thrust. Props work best under constant rpm's.

So, in your diagram, if you replace the axle with a large fly wheel, that will offer the resistance you need to safely and efficiently pump the mechanism. Add a belt drive from the wheel to the prop shaft,and that will smooth out the rpms to the prop, resulting in longer duration and smoother thrust, drastically increasing your efficiency. With a belt drive, it would be trivial to change your standard prop RPM's too. After seeing what RPM's on the flywheel you can easily maintain for the length of the race, you can then change the pulley ratios on the fly wheel and prop shaft to suitably match the optimum RPM's on the prop you select.

An old car wheel (tire and hub) should be sufficient for your purposes as a fly wheel. If you really want to make the fly wheel more efficient, fill the tire with concrete. And as it's a steel hub, you can easily weld on the axle to the hub, and the pulleys and gears from there. I'm assuming you were thinking 3d printing the gears to save weight, but if you add a flywheel (which you kind of have to), it will make weight saving gears kinda of pointless. Minus the welding, you can probably find everything you need for this at a junkyard for under $50.

And since it's a wacky boat race, I'm assuming you have to build your own hull. That's gonna be an issue with a concrete tire on board. I'd go with plywood with well sealed joints.