So I was writing an application that requires access to the registry. I had not touched any build settings, wanting to get the thing working before I added the other touches
This action can be achieved by selecting "Enable ClickOnce security settings" (since it cannot be "unchecked" during a Publish, as stated) and then by selecting "This is a partial trust application". "Local Intranet" will be automatically selected in the drop-down menu which is perfectly fine.
Save your changes, Publish the application, done-skis. :-)