Displaying Power Automate Button Trigger Timestamp in Power BI Report

I would just add a step to the Power Automate flow that logs it, personally. I think the flow run data is available in the APIs somewhere, too, but it's only stored for a short period so you'd still need to record it somewhere if you want it to be kept more than a month.

It shouldn't require manual refreshes, but you're right in that it will delay updating the info in the report until the scheduled refresh time if you store it in SharePoint since you can't direct query SharePoint with Power BI. You could use one of the other direct query-able sources like a Dataverse table or SQL to store it, but keep in mind that will be a premium Power Automate license (which is not very expensive, and you only need one for the account running the refresh... but mentioning in case that's an issue).