I think the question actually refers to the opposite situation - i.e. is it possible to set up a stack as a partial but then have an “un-pinned” background that can be changed on a per stack basis.
The answer to this question is unfortunately no. Settings cannot be unpinned in Partials and given that the background (and other child stack images) are specified in the settings panel there is no way to change them once made into a partial.
The best solution though does come back to the answers mentioned above. Just use a ProStyles layout master as a partial for the Section layout and then apply this to whichever individual stacks you like with their different backgrounds.
You can of course use Easy & Total CMS images to change backgrounds on a per stack basis even in a partial but I think this is probably beyond the scope of the original question.