Hriston
Dungeon Master of Middle-earth (He/him)
All you need to do to get events to be credible is to roll opposed Stealth/Perception checks to find out if the assassin did something to awaken the sleeper!
What would make sense is to find out whether or not the assassin did in fact make a commotion, or at least whether or not the target woke up, taking into account the Stealth skills of the assassin, the Perception skills of the sleeper, and luck.
I have two problems with your approach to finding out whether the sleeping target awakens before the start of combat.
First, sleeping creatures are unconscious. Specifically, they are unaware of their surroundings. This means their passive Perception is turned off. The latest Sage Advice covers this: http://dnd.wizards.com/articles/features/james-haeck-dd-writing. Jeremy Crawford states it explicitly at around 22:30. The sleeping creature's Perception couldn't possibly be a deciding factor in whether it wakes up or not.
Second, your willingness to engage the fortune mechanic with reference to the assassin's DEX (Stealth) check seems to assume that the task of sneaking up on a sleeping creature wouldn't be a very easy task, even for a trained assassin. IME, it's very easy to avoid the notice of a sleeping person if that's your goal. And even if you aren't trying to avoid notice and are instead trying to accomplish some task that makes a loud noise, the sleeping person still won't notice you until after s/he is first awakened by the sound. For example, if I come home from work late, and someone is asleep in an adjoining room, it's very easy for me to stay quiet and avoid awakening that person, but if I start doing the dishes because the kitchen was left a mess, it's quite possible that the noise I make will wake up the sleeper.
For fun, I thought it might be interesting to lay out the mechanical differences between our approaches. Your method results in four possible outcomes.
1. The assassin wins the DEX/WIS contest and initiative. S/he is unnoticed, and, when combat begins, the still-sleeping creature is surprised. The assassin attacks with advantage if within 5 feet of the prone creature, and any hit is critical. The sleeping creature awakens upon taking damage and remains surprised until the end of its turn.
2. The assassin wins the DEX/WIS contest, but the sleeping creature wins initiative. The assassin is unnoticed, and the sleeping creature is surprised at the start of combat, but, although it remains asleep, it ceases to be surprised at the end of its turn but can't take reactions because it's still asleep. The assassin must be within 5 feet of the prone, sleeping creature not only to gain advantage on the attack but to score a critical hit as well. Upon taking damage, the creature awakens.
3. The sleeping creature wins the DEX/WIS contest, but the assassin wins initiative. The sleeping creature notices the assassin and wakes up before combat starts. The assassin attacks the still-prone creature with advantage if within 5 feet. On its turn, the creature can stand up from prone and/or move and act.
4. The sleeping creature wins the DEX/WIS contest and initiative. It notices the assassin and wakes up before combat starts. On its turn, it can stand up from prone and/or move and act normally. The assassin's attack may have advantage if the assassin remains unseen, but otherwise the assassin attacks as normal.
My method, on the other hand, produces two outcomes that are possible.
1. The assassin wins initiative. S/he is unnoticed, and when combat begins, the creature is awakened and surprised. The assassin attacks with advantage if within 5 feet of the prone creature, and any hit is critical. The creature remains surprised until the end of its turn. (Note that this is mechanically identical to your outcome 1., the only difference being that I have the sleeping creature awaken to be surprised, which is narrative rather than mechanical.)
2. The sleeping creature wins initiative. The assassin is unnoticed, and when combat begins, the creature is awakened and surprised. The creature stops being surprised at the end of its turn and can take reactions on the assassin's turn. The assassin attacks with advantage if within 5 feet of the prone creature but doesn't auto-crit. (This is similar to your outcome 2., except that the creature gets to take reactions by virtue of having won initiative.)
The main difference in outcomes I see here doesn't stem from the timing of the creature's awakening. It stems from the fact that in my approach the sleeping creature is automatically surprised due to being unaware of its surroundings.
Was the sleeper going to wake up to empty his bladder anyway? Does a drunk smash a bottle outside and waken the sleeper just at the wrong moment?
Roll that opposed check and find out!
How does skill in Perception make someone more likely to have a full bladder? Do low WIS/low Perception characters in your games frequently wet themselves?
How does an assassin's botched DEX (Stealth) check cause a drunk to smash a bottle outside? Or is it the sleeping creature's high Perception that allows it to be roused by the constant smashing of bottles that exists in your game-worlds?