Shouldn’t the Lannisters try to become a naval power given their wealth and proximity to the Iron Islands?

They have some naval power – 25 warships isn’t nothing. And at times they have had more: there was the extended war with the Iron Islands during the Famine Winter over the mutilation of Lelia Lannister, there was Gerold Lannister’s raid of the Iron Islands to take his hundred hostages, Tommen II’s great golden fleet lost in Valyria, Tommen I’s fleet which he used to bring Fair Isle into the Kingdom of the Rock.

But something happened at some point to moderate the Lannisters’ naval ambitions, and led them to discourage Fair Isle from maintaining a navy of its own despite the Ironborn threat. Not sure what that was; it’s possible that, after the Conquest, the Lannisters thought they didn’t need a navy since they could call upon the Royal Fleet.