Creating a tooltip that is interactive but stays open until you close it

Native tooltips can't do that — they close the moment your mouse moves away, that's just how they work in Power BI.
The closest workaround is using bookmarks + a hidden panel. Basically, clicking a row shows a panel with the detail content, and an X button hides it again. Same idea as your tooltip but pinned to the page.
Only catch is it gets messy if you have lots of rows. Might be worth asking your customer if drill-through would actually work for them — it achieves the same goal and is way simpler to build.