Should I fill in the moat around my allotment?

I personally would follow your plan to create raised beds adding organic matter as mentioned as it will stop you treading down the soil on the beds but in the area where your lakes have appeared, consider planting perennial thirsty crops such as rhubarb and jerusalem artichokes, even a cranberry or two - they will love the extra moisture. 

After many years battling with different garden conditions, I have come to the conclusion it is far better to work with the soil conditions than try to change them.  The previous plotholder clearly worked hard and has made the plot more manageable, but I think it will take years and years to change this drainage situation (the plot opposite me has been fighting her flooded plot for 10 years trying to dig organic matter in to improve it, yet it floods most winters) - you are far better finding a way of working with it and using it to your advantage.