There are wifi extenders that should do transparent roaming quite well. Some can be connected using RJ45, some just take the wifi and "repeat it" further away, Some can do both.
In this case, you don't even need to connect to each.
As the article mentions in section 17, WiFi extenders are not recommended unless you don't mind slashing your speed in half. Two repeaters will slash your speed to a quarter, three to an eighth, etc. While they do extend range, they do so effectively by halving the speed because they are repeaters, which repeat every data packet they receive to get it to the other side, which wastes half of the available bandwidth. If it is only using RJ45, and not repeating the WiFi signals, then it may be OK but at that point it's acting more as an AP than a repeater. Section 18 of the article suggests an AP (eg. router in AP mode) is the best way to extend WiFi range.
I've had bad experiences with those repeaters, but never used the RJ45 backed ones. IMHO, that is then the same as an AP though - except for maybe sharing the same network SSID.
In this case, you don't even need to connect to each.