Okay...here is my watermelon growing technique:
NEVER EVER EVER start watermelons earlier than 6 weeks to plant out. Watermelon have the longest tap root of any fruit and they HATE having the tap root disturbed. Use peat cups that can go right into the ground.
Dig a hole 3 feet deep with a hand-held post hole digger or tractor auger. Fill the hole with a mixture that contains equal parts of the following:
Sand
Peat Moss
garden soil or planting mix
Allow melons to grow and set fruit and then keep the vine lengths to no more than 10 feet and no more than 3 watermelon per plant. Placing black plastic or black tar roofing shingles laid beneath ripening melons speeds up ripening (as well as trying to keep the developing fruit uncovered from the shade of the vines.)
We grew watermelon in Wyoming successfully when most other people coukd not. We even grew some of the bigger varieties like Rattlesnake, Charleston Grey, Black Diamond. We also successfully grew Moon & Stars.
We mulched each plant with straw and used soaker hose to water. They take a lot of water.
You might be able to trellis a smaller variety like Sugar baby but you would need a heck of a "sling" tied to the fence to support the weight. They do have bush varieties of watermelon and that might work better for you:
Watermelon 'Bush Sugar Baby'
Citrullus lanatus Watermelon 'Sugar Bush'
Citrullus lanatus
Hybridized by Burpee, 1977
Watermelon 'Bush Snakeskin'
Citrullus lanatus Watermelon 'Bush Jubilee'
Citrullus lanatus Watermelon 'Bush Charleston Gray'
Citrullus lanatus Watermelon 'Bush Baby II'
Citrullus lanatus Watermelon 'Bush Desert King'
Citrullus lanatus