I have some thoughts, but will make the disclaimer that I have never brewed with it before.
Best option: make a tincture/homemade extract. Soak a whole cut/cubed, even pureed watermelon in a shallow container with vodka, covered for 1-2 weeks at room temp, giving it a shake every time you walk by. I wouldn't use amazing vodka, but I certainly wouldn't use Popov either. You can run mid-grade vodka through a brita filter a few times to make it better. Once the two weeks are up, put it through a cheesecloth, and add the extract to the beer at packaging. This way you can add it to taste. Also, you get the aroma of fresh watermelon (as opposed to fermented watermelon), which is really what you are after. Also, you can package some of the 'naked' beer so you don't have a full 5 gallons of fruit beer if you dont want it.
Other option: Make the beer as normal, then add cubed/pureed watermelon to the fermenter. Drawback: with fruit, you typically want to add one pound of fruit per gallon of beer. Watermelon is very, well, watery, and may dilute your beer. Also, I would highly recommend adding the fruit in a paint strainer bag, as racking from a fermenter can be a gigantic pain with fruit pulp everywhere.
You could also do a combination of these two, adding the homemade extract to supplement.