I had a related topic about trying to replace google wifi.
Here is what I decided on after comments here and posting about it on openwrt for a 1gbps connection.
Router (~$65):
4GB version. 1GB version can’t boot with OpenWrt, it’s different thing, also don’t pick R4SE the eMMC can’t be used.
The idea with this router is to completely separate routing functions from the AP(s). This way you avoid ending up with a jack-of-all trades solution. This also makes it easier to look at APs that may not be strong routers.
APs (~$140/2APs):
wsm20 makes a good AP (preferably wired), but they aren’t all that fast as router, which should not matter in this situation.
Total Cost: ~$205 or about €190 which is the low end of your budget.