I've yet to make a watermelon myself, but its in the plans;
Like Julie noted, I would keep it as cold as your yeast strain will allow while still fermenting without issue - depends on the strain you use.
I would also start the yeast culture before you start processing the beginnings of the must. Let the yeast get rocking for some hours before you start processing the fruit so as soon as you're ready to pitch the yeast, the yeast is ready to go.
And you want the smaller watermelons, they're more flavorful. The bigger ones are blander because they're pumped full of water. The smaller ones will be a darker, deeper red on the inside where the larger ones are washed out - pretty much literally.
A good yeast nutrient schedule will go a long way towards a successful fermentation as well.
Does anyone (... psst, Julie....) know what pH/TA to go for on the Watermelon?
I cant imagine the SG was much above 1.090?