It would be legal to skin and quarter it in the field but you aren't really going to gain much by doing it.
You will still need or want to bring out the entire head, hide and paws plus all the meat. All you would be leaving would be the bones and that is not where the weight of a bear comes from. Just the head and hide are a load. Then the meat is another load. As Woods Walker said even skinning a bear in the field would be a challenge. I have skinned a dozen or so of them and I don't want to do it in the field than pack it out. I figure it would still be easier bring the whole bear out by one means or another.
The worst part of moving a bear is that they are like a big water balloon and no matter where you grab hold and start pulling all of the weight seems to immediately flow to the other end. Once you figure out how to get them hooked up for weight distribution though they aren't too bad moving. By putting a rope around all four feet I was able to drag a 350 pounder up a plank and into the back of a pickup truck by myself back when I was still in my prime. I sure wouldn't want to try it now, forty year later, though. But the point is that figuring out how to get the weight equally distributed is a big part of the challenge when moving a large bear.