Personally, I like the LFS advice. I would run the skimmer in the sump on it's own pump. I would drain the overflow with 2 lines directly into the sump. One day something is going to go down that tube, either fish, snail, or something. I wouldn't want it going into my skimmer
But hey, I like pumps, I run 5 in my sump -lol
Not knowing what type of skimmer it is, it might not even work without a pump. is it airstone driven? If not, gravity probably won't have enough pressure to work a venturi well.
Most overflows are noisy, so you need to modify the intake with a durso or stockman type fitting to get it quiet. You may not have room judging by the pics.