Rebuilding live well

[blufrog1]

New to me 85 crestliner 16' . Started checking things so I could finally take it out and found a major leak under the livewell. Pulled it out and found a broken drain hose, so I decided to rebuild the wooden lid and seat base as well as replumb it, and clean it. Well it is mostly covered with old dark blue carpet which is old and nasty so I started scraping it off and now I'm wondering it I should insulate the outside as it will be in the open in the Arizona sun. If so would 1" insulation board glued to the tank, then covered by some light colored indoor outdoor carpet be enough? Note; this is my first ever livewell. The third pic shows its location when installed.