Unfortunately, that's been my experience with WISPs too. They're fine when usage is low. But after a certain number of users, the service goes to h*ll. In our case, it was during certain hours (coincided with peak business hours) when the service was bad. During early evening and after midnight, it was stellar. That pretty much confirmed that the problem was oversubscription.
As for figuring out if the problem is the access point or the WISP's Internet connection, you might be able determine that from a traceroute. If they've got multiple hops before you get from their access point to the Internet at large, traceroute (tracert in Windows) will show you at which point the dropouts or slowdowns are occurring. But if the WISP isn't working on their own to resolve the oversubscription issue, a traceroute won't get them to work any harder. All it'll do is satisfy your curiosity.
As for your neighbor, if the WISP is running 5 GHz 802.11ac, then yes it could cause issues being that close to each other. 802.11ac is partially directional. The earlier specs (n, g, b, a) were omnidirectional so essentially everyone's signals "overlapped" if they happened to broadcast at the same time. You can test it by having one of your unplug your WISP system for an hour or so during a time when you're experiencing lag. If it fixes the problem and the ISP is unable to resolve the issue by giving you better equipment, you may actually be better off canceling service. Then get an extra router and run a cable to your neighbor's house so the two of you can share the same WISP connection, and split the bill. This probably violates the WISP's TOS, but they're already in violation for failing to deliver usable service to you.