I completely understand why it would be improper to sleep during the day on Rosh Hashanah, but if one’s davening or mood would be negatively affected due to lack of sleep, I would think it would be appropriate to sleep enough to make one feel refreshed and able to focus more.
And while it of course would be nice to learn in your free time, it would also be nice to help your spouse set the table, clean up, prepare food, take care of kids, etc.