See how it goes first without it under those conditions you may be alright, the Cozy is fairly light if not weighted. If you do need the third brake the axle stub can be changed out to adapt nearly any wheel / brake combo. It would require machining a new axle stub* for the wheel of your choice, either a hub brake with mechanical linkage or a hydraulic disc brake. Either can be Siamesed with your rear brake pedal to activate it, a disc would probably be easiest and all of this could be sourced from an MC wrecking yard.
For this you would want to farm it out to a qualified machinist and if the axle is weld on not bolt in it's a job for a certified welder. Claude Stanley of Freedom Sidecars supplied me with the stub and swing arm for my wheel I just had to weld it in.