If the router come with your Unifi package, very likely it is your router's problem. Does this happen to your other Wifi connected device?

This is a headache.

So far what I understand:
1 The 3DS WiFi works in other WiFi hotspot
2 3DS not working in your own WiFi hotspot
3 But your HP works with your own WiFi hotspot.
4 You have tried everything to make it work, such as reset etc.

If you absolutely must connect to Nintendo Network, probably you have to rely on your phone's tethering internet..... nothing wrong with your home hotspot.... probably the setting screen is not set up correctly in your 3DS?