OP here. I guess I have resolved to use inline footnotes, since the best solution here would require style-support for inspector footnotes. (I found a wish list item for that; strongly support it.)
One curious thing: while inspector footnotes don’t support styles, they still remember it. In experimenting, I converted some inline footnotes to inspector ones, and then converted them back, and the original styles (emphasis, citation, etc.) were remembered. Which suggests to me that there is at least some way that that information is being stored (but I’m no programmer).
The “unstyled italics” behavior described above still seems like a bug.