Running copper between different buildings like that is risky, because if they have different ground potential you could get current traveling through the cable and zapping the equipment. A lot of people use fiber for those runs for that reason.

If you want to solve your issue as cheaply as possible while eliminating the aforementioned risk, move AP2 to where AP1 is and replace AP2 with a PoE-powered access point. This will ensure all devices are powered from the same house, and you get rid of the failing device.