The Kindle derives its marks on the progress bar from the toc.ncx file. Edit that as you want. The rest of the epub can stay as it is.
At present, or at least the last time I checked, the Kindle only pays attention to the top level of the toc.ncx. That is, it doesn't pay attention to the lower levels of a nested toc. So you could keep the h1 chapters at the top level and the others nested beneath it and kindle will ignore them. However, this may change in the future (or it may already have as I don't have the latest Kindles).
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