It sounds like a fairly limited space you have to work within there. However, I suspect there are likely quite a few options available to you, most utilising prefabricated fibreglass lap pools. I'd imagine a fibreglass option would allow you to push the limits on how close you could build to the fence and house. If Google is anything to go off, then it suggests that lap pools can be built very close to houses and fences.
If you decide that an in-ground pool will not work for you, consider one of our above ground pools. There are a variety of rectangular shaped pools which include sand filters that would be perfect for your space.
I'll be keen to hear what experiences our helpful members have had.