I totally agree that waking up at 5am to take the tablet, then going back to sleep until 7 is ridiculous. I really struggle in the mornings to wake up, but funnily enough, have no problem taking the tablet at 5 
According to the leaflet with the Levothyroxine tablets, it says: "...take your tablets with a glass of water, preferably before breakfast."
My "research" includes:
Should you take your thyroid medication at night?
"Every thyroid patient has heard the advice that for best results, we should take our medication first thing in the morning, on an empty stomach, and wait at least 30 minutes to an hour before eating. (And also, that we should wait at least three to four hours before taking calcium or iron, which can interfere with thyroid hormone absorption.)"
It then goes on to suggest the benefits of taking it at night.
"Taking medication at bedtime instead of in the morning could have major implications for many thyroid patients.
First, it?s easier, as you don?t have to worry about when to eat breakfast.
Second, it?s easier to avoid medications, supplements and foods, like calcium, iron, and high-fiber foods that can interfere with thyroid medication absorption.
Third, coffee drinkers would not have to wait until an hour after their medication to enjoy their first cup.
Fourth, it might offer some improvement in symptoms to people who are just not getting optimal absorption by taking thyroid medication during the day."
Foods & supplements that can interfere
"In particular, iron from food or iron pills, and calcium from food or calcium supplements can keep your full dose of thyroid medicine from being absorbed.
Instead of changing your diet, the best way to keep your thyroid treatment stable is to take your thyroid pill at a time separate from your mealtime. Take your daily pill one hour before eating or at least two hours after eating. This will ensure you get your full dose."
The general consensus does seem to be that you should wait at least an hour before eating, and several hours before things like caffeine, calcium (milk) and vitamin supplements with iron in it.
We tend to eat late (after 8 pm) and I'm not sure I could guarantee I haven't eaten for 4 hours before going to be bed. However, I could make more of an effort to eat earlier so breakfast and sleep isn't interrupted.