Rather depressing to think most of your naval assets being made obsolete by a combination of fishing crates and long range ballistic missiles.
The arms race is like that, really. It's generally can be explained as 'what could kill others while keeping you in safe distance' is the penultimate goal of weapons evolution, but there's cost/performance involved in that equation. Anybody could see the cost difference between keeping a fleet afloat versus investing in guided missiles and anti-air defense systems, which doesn't need that much people to be trained or manning them.
So... I did some research and found that the JMSDF is very unlikely to name a ship like this "Akagi" and not for political reasons. See... their "helicopter destroyers" have been following the IJN's battleship naming convention and "Akagi" was the name for a either a heavy cruiser or a battlecruiser. If the JMSDF decides to resurrect Akagi, it would be a guided-missile cruiser meant to provide air defense to Carrier... Task...
MOVING ON! Honestly, if the JMSDF wants to follow their usual trend of inspiring fanarts (I'll have to admit... JS Taigei was a pretty funny in-joke) they would most likely give a ship like this the name of "Nagato" since that's the only name available that wouldn't be considered cursed.
The names "Yamato" and "Musashi" would probably be saved for nuke carriers.