Most range extenders don't require Ethernet backhaul, since "wifi over WiFi" is their main usage case. And you wouldn't need a range extender to increase the receiver gain (in dBi) at that weak spot. You could just use a wifi client (like an Alfa adapter) with a high gain antenna, say 9-12 dBi, and that should increase the RSSI (signal strength) by 9-12 dBm at that weak spot.
A range extender does have some receiver gain from its onboard antennas, but that's not primarily what it's meant for. It's meant to repeat a signal farther - maybe double the distance - but at the expense of throughput (Mbps). Unless the extender uses different wifi channels for fronthaul to the client and backhaul to the router, you'd get half the Mbps or less...