The difference comes from ease of use and accessibility. Websites usually place settings in proximity to other user options like profile, and edit options.
You can't place it in the nav, or the bottom !
I see your dilemma here, and it's good to see that you have thought upon all possible options. Here's what I think about it. One of the options that you suggested, where the filter overlaps over the main navigation seems like a good solution to me. You cited that it seems to be a bad UX, here's my argument for it.
In this case, when the filter ...