HI Markimark, the eap 225 outdoor you have asked about is a dedicated Access Point intended for a cabled connection from your router (or another device on your network) and is a great choice to extend your wireless coverage outside.
It supports POE (power over ethernet) this means using a single cable to the device you can both power it and connect it to your router.
A repeater on the other hand, which you suggest you use already extends your wifi by being both a wifi client (it connects to your existing wifi) and an Access Point (it offers connections to your other devices) without needing to be connected using an ethernet cable - it still needs power obviously.
Options;
1. can you move your repeaters to get better coverage outside?
2. does your existing repeater have an ethernet socket, some do like the RE450 or RE205 (and many others). The you could connect the EAP225 outdoor to the range externder using an ethernet cable if this is more convenient than running an ethernet cable all the way back to your router. The EAP225 outdoor will need to be configured independantly though and the performance (speed of download) will be slightly slower than a directly connected device because repeaters always add additional delay and bandwidth.
3. extend your coverage with another repeater, these also support WPS as you requested although they are not designed for outdoor use so can you position it near a window that overlooks the pool (walls attenuate the signal more than windows).
4. bite the bullet and run an ethernet cable to your router and connect it to the EAP225 outdoor, this will give the best performance, the device will need to be configured (no more complicated than setting up wifi on a router), and you can manage it as a standalone device no need for a controller for such a simple extension.