Certainly it's feasible. It's Batemans Bay, by the way, not Batesman, but I'm sure you won't get lost.
Perhaps you might want to investigate Jervis Bay, a really nice part of the world with very white sand and very clear water. You can camp there, if that interests you, otherwise it's all national park.
Lakes Entrance is a bit run down and tatty, though I don't know of any particularly great towns in that area anyway. It's fine, but it's not as nice as the towns on the NSW coast.
Wollongong isn't necessarily a place you'd stop as a tourist, it's a fairly dull medium-sized industrial city.
The Royal National Park will add a bit of extra time to your journey. It's a nice national park but it's not that different to a lot of the stuff you're going to be unavoidably seeing further down the coast, so perhaps you could omit it.
By May it'll be getting a bit chilly so don't expect to be swimming at all these beaches. Definitely not as you get further south.