I doubt a properly maintained pool with filter and chlorine, properly shocked, would be hospitable enough of an environment for mosquitos. Your neigbor's standing water, however, is a concern.
I'm looking at this gardening catalogue I have next to me that has "mosquito dunks" in it. These are compressed disks of bacillus thuringiensis subspecies israelensis. This is a bacteria that is deadly to mosquito larva, but harmless to humans, animals, and fish. Each disk lasts 30 days and treats 100 square feet of surface area. This catalogue has them costing $13 for 6 disks. Maybe you can find something like these and toss them in your neighbors' standing water or ask him to do it - your treat.
I haven't used them, but they might be worth a shot. I have, however, used this spray by cutter that hooks onto the hose and sprays over grass, concrete, whatever. It is harmless once it dries, and has worked remarkably well on my lawn. It says you can spray it on veggies as well, but I like to keep my garden pesticide free as much as I can. This supposedly works for a month even with a lot of rain. I need to see how well it holds up since we are under heavy storms right now in DC and it's been on there a week.
Maybe a combination of these low tech approaches might give you some relief before you shell out that much for a machine. I thought about it too, but decided to see if some of these less harmful chemicals and bacteria would work. I need to get those disks and toss them in my neighbors' trash cans. He insists on saving rain water to do God knows what with.
Maybe try a few of these first to see? I know that Asian Tiger mosquitos are out and are voracious biters. They don't care about the heavy midday sun either, which is bad news. Even worse is the fact that in my area they carry the West Nile virus. I've been able to get them down pretty low with the cutter spray.
I would suggest you also look for all areas of standing water in your yard. The above mentioned mosquitos can breed in a bottlecap worth of water. That's not a lot at all, and small water deposits like this could often be overlooked.