Unless they've had a recent change, Anna & Elsa are only available until a half hour before regular park closing time and don't stay through evening EMH.
So for example if the regular park close is at midnight with EMH until 2 am, A&E are still only available until 11:30 pm.
On a recent visit, my kids weren't interested but I did talk to a family that said they waited about 10 minutes to meet them just before 11:30. The posted wait time was 90 mins.
As with just about anything else at Disney, the crowds die down in the evenings and wait times become very reasonable. Keep in mind that the posted wait times can be greatly exaggerated.
I like your plan to skip the morning madness.