First, make the pickled onion. Put the lime juice, vinegar, sugar, bay leaf, halved jalapeño and bay leaf in a food storage container that has a tight-fitting lid (a glass one is best). Stir until the sugar is dissolved. Then peel the onion, slice it very thinly, and add to the container. Pop on the lid and store the container in the fridge for at least an hour—this will be more pickley if you allow at least a day.
Next, peel the watermelon and cut into small cubes. Put in a bowl or container, cover tightly, and refrigerate to make it nice and cold.
When you are ready to assemble the salad, make the dressing by mixing together all the components until the sugars are dissolved. Taste it—you may want to add more sugar or lime juice.
To assemble the salad, put the watermelon cubes in a bowl. Tear or cut the mint and Thai basil into little pieces and scatter over the salad. Add the dressing—start with about half and gently stir into the watermelon. You want it to coat everything and for only a little to pool in the bottom. Then add the pickled onion—again, start with about half and see how that looks (discard the pickling ingredients). Stir again gently, then taste. You may want to adjust with a bit of white sugar or a little salt.
You'll notice that the dressing and the pickled onion don't have any salt, because the fish sauce brings plenty, but at this point in the process, you may decide to add just a dash of salt if you think the salad needs a bit more liveliness.
And that's it. Serve while it's still cold. You can also refrigerate and serve later the same day.