Image in docx with wrap set to "behind text" works fine in microsoft but broken in libre

Of course, but users rarely customise built-in styles (all the more for Word switchers).
Header is already configured for “complex” headers: one block flushed left, a center block and one block flushed right. Usually one of the side blocks contains the page number, the other one some fixed title and the center block the chapter title. Effectively, this doesn’t really corresponds to OP’s use case.