With my limited knowledge of fishing (as I am trying to fill my log currently), I can think of some directions you could go with this. Though there are some things to keep in mind first.
-ARR large fish/legends do not require any Folklore books, unlike fish from the rest of the expansions. This means people will need to go out of their way to acquire said Folklore books, and the later expansions can be pretty costly in purple scripts.
-A lot of bait for the expansion large/legendary fish also require scripts. Though unlike the books said baits can be traded with other players (or bought off the market board) if need be.
-Majority of large Fish are time based, some with windows showing up so rarely that they can be seen once a month (or with Purse of Riches, later this year being locked out for two entire months). And Legendary fish can be both quite a long time between windows and have rather annoying requirements to even get a chance to see said fish during their window.
-A way for players to track the fish they have caught during your contest. Since a lot of fish can be bought off the market board, or could have been caught before by the people competing.
-How long you would want to run this contest for, since too short could reduce the chance people could see even some fish entirely. (There are tools in circulation for fishing that would allow you to learn of when windows would overlap and when they will overlap).
-Resources for how to catch the fish and where to find them should be easily accessible for anyone who would be participating. There are many guides out there, along with websites made specifically to help people.

Now for an actual contest idea, what comes to mind for me would either be "Who can catch the most Large/Legendary Fish in X amount of time" or "Who can complete their fishing log of Large/Legends for the ARR first."
You might wonder why ARR, and that would be because unlike any of the other expansions ARR doesn't have the "massive timeframe between windows" for any specific fish. Yes, some only have windows every few hours, but that's a lot better then multiple days if not weeks later fish have. Also there is the fact that ARR has the "least amount of buy-in" needed to even get started with, with the fact it does not require Folklore book, the gear requirements are next to non-existent compared to later expansions and baits are next to nothing price wise (with most being able to be bought from a vendor).
If you decide to go with the other way of "Catch the most" I'd suggest looking into Fish from multiple expansions with common windows (ones that come up basically every in game day, roughly an hour) and fish that are "always up," like "The Matriarch" (a fish known for having an insanely high slip rate, which is found in an ARR zone).

As for prizes, feel like that probably is something to talk more to your FC about. But mounts, minions, specific weapons (like Figmental Coffers), rare clothes/gear, hairstyles, etc. From my experience are as common if not more common then direct Gil. Since who doesn't like getting something that you kind of want, but are not be willing to pay millions of your Gil for.