TheBritishAreComing :
Before you go spending money on a solution that's not fit for purpose, as a professional installer of long-range wifi solutions, I thought I might be able to offer a voice of reason here.
By placing a higher powered wireless access point at the router end of your connection, you are only fixing half of the problem. Chances are the device you are using at the pool (whether a smartphone, tablet or laptop), still only has a limited range (probably in the order of 200ft on a very good day), so while it may be able to see the wireless network, it's not going to be able to connect to it.
Unfortunately, if you want to get a wireless signal to travel 2000ft, the only way to do this is to install a pair of high-powered devices, one at the router and one at the pool; then point them at each other to form a wireless bridge. Only by doing this will both devices have sufficient transmit power for the signals in both directions to reach each other.
Chances are this is going to be out of budget, not least because it will also require a suitable provision for a power supply at the pool end.
Sorry to have to be the harsh voice of reason on this occasion.
The best solution in fact would be to run a fiber-optic cable from the house to the pool, connect this to the router (you'll need a media convertor to convert it back to ethernet at each end) and connect it to a basic (reasonably priced) wireless access point at the pool. However this also will need power provisioning to the poolside.
TheBritish-
You are, of course, correct in regards to the rf power part of the problem, and that the BEST solution is to run a fiber optic link out to the pool. However, fiber may not be cost-effective (the cost of trenching, running conduit, pulling fiber, installing the hardware, and then re-covering the trench), so wireless is the best alternative.
You have failed to consider the effects of antenna gain - the AyrMesh Hub and Ubiquiti PicoStation use 6 dB omnidirectional gain antennas, and gain antennas increase both the outgoing AND the incoming signals. So a low-power device like a cellphone or a laptop can gain a LOT of range by connecting to an access point with a high-gain antenna.
Interestingly, this trick only really works up to about a 9 dB antenna. Omnidirectional antennas achieve "gain" by directing less of the signal up and down from the antenna and "squeezing" that signal into a tighter "donut" around the antenna. When you get above 9 dB of gain, however, the ring of signal around the antenna is so thin that there's a relatively small region that gets the full signal strength. These are results that are not obvious from mere "book-study" of the technology - my knowledge has been significantly enhanced by installing a lot of outdoor WiFi systems.
The advantage of a meshing system like AyrMesh (or Meraki, Aruba, or Open-Mesh) is that you can have chain out several access points to increase the range of the system, albeit at the expense of bandwidth. The advantage of a high-power access point like AyrMesh over "normal" 100 mw access points is that the access points can be quite far apart, because of their high power radios and high-gain antennas - up to 2.5 miles in the case of AyrMesh.
So you can put one meshing AP at the house and have another at the pool. You may not have hundreds of Mbps of bandwidth, but you will have very strong signal at the pool and enough bandwidth for most applications, and the cost (a few hundred dollars) will be MUCH less than the cost of running fiber.
You can also, of course, have a technician install a high-bandwidth Ubiquiti (or other brand) point-to-point wireless bridge and put a normal WiFi Access Point at the pool - Ayrstone (the company behind AyrMesh) has a very easy-to-install bridge product for up to 5 miles, and Ubiquiti has a range of bridge radios that will provide varying levels of bandwidth at varying levels of price. That will provide much higher bandwidth at the pool, but, for the Ubiquiti radios, you will probably need an experienced technician to install the wireless bridge. A wireless bridge also won't provide WiFi between the house and the pool the way a meshing WiFi system will.
Bottom line (at least from my perspective): there are a lot of ways to get WiFi out to the pool; it matters whether you want to have massive bandwidth (remember, most rural Americans are lucky to get a few Mbps of Internet service, so bandwidth is not always an issue) and whether you want to have access in the area between the house and the pool.