A hose end sprayer would be more efficient but a carry around sprayer will work. You want to use something that sprays evenly like a dial a flow rather than a miracle grow type sprayer that is more concentrated in the beginning. What ever system you use, you should stop and shake the container ( use the golf ball trick and put a golf ball in the container to help keep agitate the solution) every 25 ft or so. The use of food dye helps since it gives you a better idea of how well you have covered the space. Since you are mixing the nematodes with water to apply them, you should not have to water them in the next day. They are living organisms and should burrow down by themselves. You do want to get even application to cover the area. The most important thing to remember when applying live organisms is to make sure the environment they are going to is healthy for them. I would water the lawn before and aerate it to make sure the soil is not rock hard to get into. Watering well will also bring the grubs to the surface. It is what we did as kids when we used to go grub hunting. We would have our pail and weeders and flood an area of the lawn around the edges of the brown spots. The grubs don't want to drown so they come up to the surface and we picked them off with chopsticks and dropped them in the pail of soapy water. The soil temperature needs to be above 42 degrees or they won't live long enough to get rid of the grubs. I woud wait a day then apply the nematodes and water them in immediately and not wait till the next morning. You want to make sure the nematodes get down into the soil where they will do the most good and not just be on the surface.
Happy gardening in Hawaii. Gardens are where people grow.