Ok, so I'm embarrassed about the level of hack in this solution, but it solves a business problem and I thought others might be running into the same issue.
Currently, the only integration between MS Flow and Power BI is via the Alerts trigger. This is handy for certain use cases, but doesn't help when you need to trigger a business process based on a human decision. It also doesn't allow you to pass a complex data object out of Power BI to the MS Flow workflow.
My workaround for this is to create a measure in Power BI that assembles a JSON object containing the required data elements I need from the Power BI report, and expose the JSON object in a user-selectable visualization (e.g,. a single-celled Table).
The report user then configures the report to get the desired result, copies the resulting JSON object from the PBI measure, and pastes the JSON into a dedicated Microsoft Teams channel. I can then run an MS Flow using the Microsoft Teams 'new item in channel' event trigger to grab the JSON object and perform operations against it.
Hopefully, at some point in the future there will be other trigger options from Power BI, but this very hacky approach has allowed us to get some integration work moving in the short term while we wait for a better solution.