We had another customer who built a one wheel drive beach wagon and had the exact same problem so he switched it over to two wheel drive which completely solved the problem. Due to this we are no longer recommending one wheel drive powertrains for beach wagons and off-road carts.
I do not think that changing to a different tire would help very much. A second motor (along with a second controller, chain, sprocket, freewheel, and adapter) could be added to the undriven wheel however that would require a lot of parts and also the battery pack size would need to be doubled with a second motor in order for the wagon to have the same drive time that it has right now, so I like your idea about welding a live axle between the two wheels so both of them are powered with the motor that is already installed.
For a 24 Volt 350 Watt kit we have controllers with reverse that could replace the existing unidirectional controller. A switch would need to be installed in conjunction with the reversing controller to control when it is in reverse mode. This switch could be handlebar mount or panel mount.
Reverse will not work with a freewheel in the drivetrain so the freewheel would need to be welded together so it is solid and welded to its adapter hub, or the sprocket could be switched over to mount to a solid hub.
I do not know what size tires will and will not work on a beach wagon. Most electric beach wagons that I have seen have wide tires though so I think that would be the right direction to go. Maybe someone else here who has more experience than I do with beach wagon tire sizes could comment on this.
Since the existing wheels have bearings they will not work with a live axle in their stock form. Also when switching over to a live axle the diameter of the axle will most likely be larger than the existing fixed axle so it will not fit inside of the existing wheels bearings.
The best way to approach this would probably be to think in terms of replacing the existing rear end with a new axle, bearings, bearing hangers, sprocket hub, and wheels. This way all of these parts could be ordered together so they are all compatible with each other. A go kart parts store is usually the best place to source parts like these.
Please let us know if you have any questions.
Nick, I'm in the middle of an axle conversion right now. I was able to source all the parts I would need from azusaparts.com. They also cut custom teeth axle sprockets and can drill out mounting holes that match their uni and mini hubs. Here's a clip of my receipt for some of the conversion parts. Hope this will help on your conversion.
1030 - 4" WHEEL, STEEL, ASSEMBLY with 1" Live Axle Hub, 2 piece Rim, 2-3/4" Wide
2562 - MINI-HUB, STEEL, ZINC PLATED, 1" BORE WITH HARDWARE KIT
8403 - KEYSTOCK, 1/4" X 1/4" X 12"
I'm still struggling with understanding how a freewheel sprocket can work with a live axle. I've purchased the free wheel and internal ratchet system. If I mount the free wheel/ratchet to the sprocket and then use a keyway/key in the axle to lock everything together, will the drive sprocket chain rotate the driven sprocket forward (live axle) and still allow me to roll the cart backward with no resistance from the motor?