A couple times I was able to traverse the KI road in winter, its having been plowed and compacted by logging trucks. Since the AMC took over the land, the road, at least at the Greenville end, is apparently closed less than a mile from the gate with a parking area plowed. The KI road from there is closed to traffic except for snowmobiles and I presume non-motorized access.
It's a shame, in my opinion, because I enjoyed a portion of Gulf Hagas in winter using xc skis and this trip is no longer reasonably accessible to me without staying at Little Lyford Pond. I had also looked forward to a couple of loops which included portions of the AT and, similarly, they are no longer accessible.
Little Lyford Pond is indeed pricey but considering its location and the inclusion of meals, its worth consideration. If you have a group and would like to do your own cooking I suggest staying at the "nearby" AMC cabin at Moose Point. That is more affordable, divided up amongst a group. Technically it's capacity is 6 I think but I recall that, if allowed, you could easily enjoy the cabin with 8.
Can't suggest any specific tentsites but would caution that some of the most dramatic scenery in the Gulf could be extremely hazardous in snow and ice.