Prize Catch & Perception Comparison
Spreadsheet documenting the individual sizes caught at 3200 perception and 4200 perception.

This type of comparison was done at the start of Endwalker on Echinos while simultaneously testing white scrip farming methods. That data has since been lost, and needed to be redone.
Moxutural Gar itself is a curiosity because it is a fish that requires Big-game Fishing to guarantee T3, meaning there would be incentive to increase perception if it provided a more narrow, higher size range but it does not appear to.
Significant work testing and comparing multiple white rarity fish with soft breakpoints was done in Endwalker and has not been replicated on a great many fish.
This has come up as a question as Mirfa Seiten@Masamune noticed a potential difference in ranges when using Prize Catch, without Big-game Fishing, with increased perception on Topiary Serpent, documented here.

The dataset is small enough that it is not conclusive, though there has been little documentation of Prize Catch on fish with hard breakpoints (which Topiary Serpent is not) outside of Phyllinos, or on ones that are otherwise more 'unusual' (which Topiary Serpent can probably qualify as).
Since gathering for Double/Triple Hook does not appear to make an exception for the Master mission it would be most relevant to, I am immediately skeptical that perception does something for Prize Catch that has not otherwise been documented.
I do personally believe that if we were to encounter novel interactions with stats and abilities, it would be on the ones that interact with perception than with gathering. There are more abilities that interact with perception to begin with and Prize Catch is [as far as we know] very straightforward otherwise.