I am sure that you are aware that your testosterone levels can go down due to sleep deprivation. Also most bodies naturally try to store fat. So gaining weight should be easy, but it might not be lean weight.
I would suggest the following:
- Get your lifting done in 30-35 mins or less each day. Your body is already tired. No use making it over-fatigued. Anything you are doing after this is not helping you gain muscle.
- Each day start with a core movement - for a good amount of weight. At the same time try to be done with the core movement in 10-15 mins. Being fatigued you may only have 10-20 mins of good lifting time. Use that for squats/deadlifts, pullups, bench, curls/skullcrushers.
- After doing the core movement I would then go to the auxiliary lifts with speed not weight in mind. Get your body shocked during the core lift(s). Afterward you should be lifting almost non-stop. You can try supersetting 2-3 lifts in rotation. Key being form and intensity - not weight. The reasoning behind this is that you need to fight your body's natural attempt to put on fat.
- Try working out in the morning. Even one day a week. You can probably more efficiently get to sleep at night if you workout in the morning.
- 4-day split (legs, back, arms, chest). I would also do a few mins of core/ab work daily - fight the fat gain.
- Sleep as much as possible on weekends. I worked night shift at one time and averaged 5-6 hours a day of sleep during the week. Weekends I would grab 12 each day and I was fine. If I missed the 24 hours of weekend sleep I felt it within a week or two. Your body can recover with extra sleep.
- Drink more water. 200 ounces of fluids a day. You are much more prone to pulling a muscle when fatigued or distracted. Don't let dehydration factor into that.
- See if you can switch one or two of your workouts to the weekend. If you only had 2 days of low sleep that is optimal. Even switching 1 would be a big help.
- Light cardio. You are sleeping less so more prone to lactic acid build up. Start a light cardio routine 15-20 mins daily. You are not training for a marathon here!
(So total workout time is 45-55 mins x 4 days a week).
I am not sure on recommending squats (given that you feel they are most important - I do too) to Fridays. You would have optimal time to sleep after but you would also be at your most sleep deprived on Friday night. Personally I would do them on the weekend or Monday. I want to be awake as possible when I have a lot of weight on my back.
I guess to sum this up. Do things to sleep more. When you are at the gym quickly do heavy weights. Use your gym time more efficiently. When you are fatigued at the gym (after doing a core lift) do not lift heavy. Incorporate some body-building theories into your routine given that your body will be more apt to put on fat.