This sounds more problematic than it has to be, imo. My bog is gravity fed (now) but I have the force coming from 40" up, not at the same level. This is so I GET UPFLOW, which is far superior to a surface/downflow through your bog media. I think in your case it would be better to pump from the skimmer bottom then into the bog (force fed through flex pvc -- larger, 4" drain tile tube with slits. The water then is forced up through your bog, which should then pour back into your pond. This will suppose you have a backup plan for if your skimmer water level gets too low and forces your pump to suck air. To get gravity fed, you need to help create the force with which to make it work.
In my case, I don't have the skimmer but all the rest above is in place, with the addition of a mechanical filter as a pre-bog filter (which is what your skimmer will do for you). I don't worry re water level going low as I don't have the skimmer. To clarify; pump in pond, water to mech filter (40" higher than bog), which pours down a tube which runs beneath the bog media, water rises up through and then over pours into my stream, then down into the pond.
From talking to Meyer last year, I also had worries I would not have enough 'force' from just gravity (bog USED to be pump fed, forced) but he assured me the 40" height would do it (and it did). There won't be much gravity force if you don't elevate somewhat, hence putting a pump in your skimmer and pumping from there instead. You won't need this additional height then.
I was unclear exactly where you were going to put your pump but if you had ideas of beneath the bog and pulling water, imagine trying to clean/fix/replace the pump. Maybe I just didn't understand but making sure all angles are covered.
Michael