"does anyone know if it is possible/allowed to cycle through there"
I would guess the short answer would be no, I doubt it would be "allowed" in the sense someone may object, in the access rights respect not sure would be allowed either. Is it possible I would think so on mountain bike, on a road bike well what is possible.
I went past that entrance today would imagine people may object. It was actually the path from the Wang Lang to extend Long Dalmahoy that I went ride. There is a closed gate saying access only on that route , think that route may be considered legal, however I did not proceed. A short cut to raf Kirknewton which has no access rights official secrets act-) also a short cut to the Lang Whang but only go that way when going south Moffatt Carlisle and would not want to use a bumpy track.
The problem with the route from kirknewton house is 2 large houses, the natural boundary of the property has no country side access rights. If a large house a natural large boundary. Also in the context of the area people may object. The rights of ways don't seem particularly welcoming. Just past the quarry on the Long Dalmahoy road there is a right of way path to Balerno with a locked gate although there is a style so could walk it.
It may be the proximity to Edinburgh perceived security issues, also land ownership if one lord owns all the land better than smaller blocks for access rights as people think a field is their private garden etc, more likely to get told off rightly or wrongly. In more rural areas borders etc much less likely to get closed, locked gates or people objecting.
I was questioned previously for "looking at someone's land" when walking down Long Dalmahoy I had just stepped of the road as a car coming in both directions the car that stop to question. I would tend to think in context that the route from kirknewton house would possibly not be allowed or "welcome" but just a guess of course if someone has actually tried the route would have better idea.