Selichos and sleep (or lack of it) - Updated

This week and next are two of the toughest weeks for me of the year. I don't get that much sleep normally and with selichos a half an hour before davening my sleep time is cut even further. I have been davening vasikin for a while and while this week is tough (selichos starting around 5:50) next week is insane. After we change the clock netz is going to be around 5:40 with selichos starting around 4:50.

What is even more depressing is that even getting up this early I am not saying selichos at the real proper time. In fact, there are very few people who actually say selichos at the proper time. The shulchan aruch in siman 581 states that the minhag is to get up early in the morning before alos hashachar and say selichos.

The Mateh Efraim discusses saying selichos after Alos and says that b'dieved this is fine and that it is better to say selichos after Alos then not say them at all.

In other words, the proper time to say selichos is before Alos Hashachar, saying selichos after Alos Hashachar is only b'dieved. Considering that next week in Israel Alos Hashachar is before 4:30AM (depending on how exactly you figure it), there will be very few people saying selichos before Alos.

It is interesting how it has become completely acceptable to say selichos not only after Alos but after netz and no one says boo about it. Why by selichos are we so accepting of doing things in a b'dieved way?

Update

One of the comments mentions what has become a common practice, saying selichos in the first half of the night. While it certainly appeals to my sense of sleep, all the sources that I saw do not recommend it (to say the least).

Here are some sources in the acharonim discussing the issue:

Mishnah Berurah 565:12, selichos should not be recited before midnight.

Sha'arei Teshuvah 581:1 quoting Birkei Yosef, one who finds himself in a shul where selichos are being recited before midnight, should not recite the Thirteen Attributes along with the congregation.

Igros Moshe O.C. 2:105, R' Moshe is very against the practice of saying selichos before midnight.

Yechave Da'as 1:46, prohibits saying selichos before midnight, instead he advises reciting selichos before Minchah.

R' Willig told us that based on the above, that instead of saying selichos at 10:30PM, a better choice is to daven mincha a little early and say them between mincha and maariv at the end of the day before sunset.