If you're going to run an ethernet backhaul I'd purchase a few less expensive wireless access points (or wifi routers with the DHCP server turned off) and a good router. To my knowledge the benefit of a mesh network is that the access points communicate with each other wirelessly, if you are connecting wires between the access points you no longer have a mesh network.
I use a Ubiquiti Edgegrouter X and a Unifi AP-AC Lite access point, coverage in my small single level home is adequate with a single access point, adding another access point would be trivial. With an Iphone 8 right next to the access point I can get ~450mbps, before I got gigabit internet, I was able to max out my 100mpbs connection anywhere in the house. The Edgerouter x is currently going for about $60 and a AP-AC Lite is about $80, assuming you needed 2 access points you'd be looking at around $220, so it's a bit more than your budget.
Pros of Ubiquiti gear is it has been more stable than consumer grade routers I've used in the past, I haven't had to reboot the equipment in 8 months. It's not the fastest or strongest wifi signal, but I've had 15-20 devices connected at once all streaming music or video and haven't noticed any problems, other routers choke with that many clients. The Edgerouter is nearly enterprise grade, very flexible, will do POE on one port (but not the 24v POE the AP-AC Lite needs,) can be configured for almost any need.
Cons are cost, although you get quality with the cost. The Edgerouter X is missing some of the features of the Unifi routers, things I consider gimmicks, but the access points integrate more seamlessly with a Unifi router than with an Edgerouter, if you want to use all the features of the Unifi access points you have to double or triple your budget for a Unifi router. The access points are POE, which is cool but it's not a standard voltage, so you need a special switch or you have to use an included POE injector which kind of defeats the purpose, but isn't a deal breaker for me. Because the router software is very flexible, it's also somewhat challenging to configure, certain things have to be done with the command line, enabling hardware NAT offloading for example, which I consider a critical feature that should be enabled by default as the device isn't capable of gigabit speeds with the feature turned off, has to be enabled manually using the command line.