use a switch from the wall plate then plug the access point into that.

Single chain WiFi = slow. Lines up with 100 Mb ethernet.

If you want to try the Edimax, there's one on Trademe for $40. Not clear if the four physical ports definitely work when in AP mode though, you would need to double check first. It seems that device can be used as either an AP or router, and those ports may only work in router mode. The discontinued Apple Airport range used to do what you want in AP mode.

Out of the box, the EDIMAX is configured in AP mode but as previously mentioned, it would be interesting if the ethernet ports work still work when in this mode.

@peejayw has this router and used it to extend their WiFi coverage. They might be able to tell us more about it.

Just read in the manual that it looks like ethernet could still work in AP mode (emphasis my own);

Access Point Mode

The device connects to an existing router via
Ethernet cable and provides 2.4GHz and/or 5GHz
Internet (wireless and Ethernet) access for your
network devices.

I have never had an issue using a wireless router to do exactly what you require. Just set it to a unique static IP on the same subnet as your router, disable the WAN (and optionally bridge it to the LAN - most third party firmware will let you do this to gain an extra port), turn off the DHCP server, set wifi to AP mode, and you're away.

Not as "neat" as a Unifi Pro but a hell of a lot cheaper.

Looks like the Edimax supports OpenWRT so if the stock firmware doesn't do the job, I'd try flashing that.

My pick would be the UAP-AC-IW-PRO which will replace your wall socket and give you 3 ports one of which is PoE

Only thing is you need stuff like controllers to make it work