Let's say that the leveling drum has sounded and the level icon has appeared. I sleep for 24 hours and see that the level up screen shows 5 5 3 as the best option. Now let's say that I didn't like that and loaded the save before bed. Is it possible for me to, with the level icon clearly visible the whole time, continue raising whatever skill so that I can sleep at a later time to recieve the 5 5 5 option?

no

No. The attribute multipliers are set in stone whenever you gain enough increases for a level up. Any further increases in skills will count for the next level up.