What I finally did was actually turn off the service, rather than using the service's settings to keep it quiet. When I used the services "active time" scheduling configuration it didn't actually turn off the service completely. Using the Query Performance Insight menu option and digging into the actual queries being run (including the actual query text) revealed the service was still querying the DB even though I had configured it to go to "sleep" using the service's own scheduler. Query performance insight was extremely helpful in discovering what was going on there.
I was able to schedule the services in question to actually shut down using task scheduler which allowed the DB to pause.
A word of warning though. I later found that navigating to and running queries in the Query Performance Insights tab actually woke up a paused database, so you want to be judicious when using that tool.