Any of the above would be fine, but once it's dead you need to get the guts & gills out.

Instead of using lake water to cool, use salt.  My thinking is that lake water isn't particularly clean or cool for that kind of storage.  A hefty dose of salt, rub the whole gutted fish with it, will work better.  Then a good rince, LNT-nutty up on shore dump the rince water in the firepit if you're burning wood that night - otherwise treat it like greywater.

Me?  I'd let it go.  I don't need the fish. I bring food.  Fresh fish is an awsome luxury.  The closer to living the better.  I can get old fish at the market.