Better not to look for what does not exist ... day rooms at Atlantis.
Book the Comfort Suites (there's no Day's Inn on Paradise Island) for one "night". Right next door and much much cheaper (around $130-150 / night when you add taxes, etc.). You can do it online.
You get a voucher which is good for charging meals, etc. all over Atlantis and includes the right to wrist bands and towels. It's valid as soon as you arrive/check in.
Enjoy Atlantis all day then go to your room shower and freshen up, check out and head back to the ship.
Not a bad deal for a family considering it would be difficult to "exhaust" Atlantis in a day.
Atlantis, by the way, is very jealous of its "upscale" status and you won't find walk-up "deals" no matter how hard you try.