Full width background

In your simpler solution the image is clipped a bit more, it’s only 0.27 mm but enough to show up as white line. if you consider the header already as frame, you’re right about 2 frames, but inside the header I add only one frame. You’re also right that marking / choosing is hard with full-size-frame in header with image and text additionally, I just never found a way to circumvent the frame.
Your mention of caption is interesting and might make it seemingly more complicated but I think it could be easily applied inside the frame or just in the header-text while latter one wouldn’t be semantically correct.

Very strange, I don’t meet this difficulty. Can you attach to your question or answer a sample file reduced to one page (no text or lorem ipsum) with the header and image for further analysis?

In case this is relevant, what is your OS and LO version?