You need a router or a proxy. A proxy would be annoying, so a router is preferred.
If you don't have control of your edge router, just get a cheap Pi-type device, install OpenWRT, setup your VPN connections, then create a route on your network to point at this new device for whatever you need it for.
If you simply want to use it at-will for certain things, you can put a proxy on it.
As to your other issues, it sounds like your WG connection is just dropping, in which case it won't automatically reconnect by default. OpenWRT has plugins that can monitor that and reconnect when it drops, or you can script it pretty quickly as well.