Ok. So the only way to get different sizes of text in each text frame, with Fit Text, is to break the text thread between the frames, and only then run Fit Text.
We have an advantage here, though, that each frame gets only one paragraph (with a style applied called "Quote Screens".)
So what I would do is:
(1) Set the Quote Screens style to start in a new frame. That's an option in the paragraph style definition.
(2) Reduce the font size of the Quote Screens to something very small, say 5pt.
Now we have a situation where each, entire paragraph is in its own frame, so we're ready to break the text threads.
(3) Run the "Break Text Thread" script that comes with InDesign.
(4) Set the font size of the Quote Screens style to 130pt.
(5) Finally, run Fit Text, choosing the "Only shrink text to fit, never enlarge" option.
You should now find that the text of each proverb fits in the frame, with a maximum size of 130pt.
You should probably keep a copy of the original version of the document before all these manipulations, so that it can be easily edited, or so that you can revert to it in case something goes wrong.