Inkscape - Exclusion/Difference: remove fill but preserve stroke (size)

For my background, I'm not entirely comfortable with Inkscape, but I'm not entirely new with it either.

Say I have a large object with a smaller object inside of it. I want to remove the smaller object with Path -> Exclusion (or Difference). If I do this, the smaller object's fill will be removed and its stroke will be preserved. However, the stroke size will be reset. Is there a way for me to preserve the stroke size when using Exclusion?

Neither stroke to path or object to path do what I'd like.

Let me know if I need to explain myself more.