Yes, you could have a linked heading that displays instead of the rest of the categories box for certain ez-pages, requiring only stylesheet work. It would take a bunch of hands-on fiddling to get it right, but not a horrendous job.
I would probably need to look at the site live to understand what is not working with the flyouts. Seeing your defines would also help. I suspect the flyouts may depend somehow on parents that are hidden, so cannot be hovered on to get the lower levels, or maybe have different identifiers than you would expect so your CSS is not catching them. Not sure until I see it in action...
Maybe the easiest way around your five-word category name display is to make an image of the text laid out exactly as you want, and use the standard feature to replace the text with that. It would have an alt/title tag of the actual name, so would still be SE/screenreader accessible.
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