@Ondulado 

Hi,

Since all three routers are of the same model TL-WR841N you should be able to single out where the problem lies by interchanging the routers with each other.

So, Router 2 and Router 3 are both set to "Access Point" mode" and are configured similarly, with basically the only differences being that the Static LAN IP's are different (192.168.0.2 vs. 192.168.0.3) and maybe different SSID names for the Wi-Fi?

Is it always Router 3 that looses the Internet connection?

If so, then try what happens if you swap Router 2 and Router 3 with each other, without making any changes to their settings.

I mean just unplug Router 2 and bring it to the place where Router 3 currently is located and plug it into that cable and then bring Router 3 to the place where Router 2 originally was and connect it there.

If it turned out that now Router 2 loses the Internet connection, then this could point to a cable issue. You could verify that by connecting this router to Router 1 by using another cable. (just bring Router 2 to where Router 1 is located and use any additional short Ethernet cable you have available) 

If after swapping Router 2 and Router 3 with each other it is still Router 3 that loses the Internet connection, then swap Router 3 with Router 1. This will require more effort, as you have to re-configure Router 3 for use as the main router and re-configure Router 1 to become an access point. But it could help you in finding out whether "Router 3" may have some sort of fault.