web: Reload after toggling the Campaigns flag

Created by: LawnGnome

This implements the approach suggested by @felixfbecker in #11118: instead of having the server tell us when a reload is required, let's just parse and pull apart the before and after configuration blobs and figure out if we need to reload from that.

My assumption is that we'll revert this PR in full as part of removing the feature flag, when the time comes.

Fixes #10715.