Tony, the loop you are asking about could be done entirely on state land. The northwest corner is unfortunately not on the NYNJTC maps, however, it is shown on a large fold-out map included in Michael Kudish's book
The Catskill Forest: A History. There is a swath of state land that comes down to the Spruceton Road, County Route (CR) 6. On the west the swath is about 0.4 miles east of State Route (SR) 42, and extends to about 0.6 miles east of SR 42. The eastern boundary appears to be at a bridge over a side stream flowing down from the south, and a short distance before a bridge over the main West Kill stream on CR 6. I have never used that swath of land, nor do I recall if there are any convenient parking spots. After fording the West Kill a straight line up the ridge would take you to a point about 0.5 mile west of the 3100' summit of Packsaddle (aka Lexington). There is a jagged line of state land, at times narrow, and sometimes a little off the ridge top that leads to Pine Island and Evergreen. The summit of Pine Island may be a very short distance north of the jagged strip of state land.
The southwest end of the loop is all in the large West Kill Wilderness Area. As Jay mentions you can bushwhack between the Devil's Path and SR 42, over North Dome, Sherrill, and the 3340' Balsam (which is not on the 3K list, only 120' prominence). To answer Jay's question, yes, the 3408' NE Halcott (historically known as Sleeping Lion Mt) is on the 3K list.
The Catskill Range names are probably not official as they (with the exception of the "Blackhead Mountains") do not appear on the USGS quads. Kudish's book names all of the Catskill ranges. Often a range is named for the highest mountain, but there are many exceptions, e.g. Lexington rather than Rusk Range. Kudish uses the term Hunter Mountain Range to cover the ridge beginning with Overlook, Plattekill, all the mountains of the Devil's Path, continuing to Sherrill. By extension that would include the 3340' Balsam, a false summit of Sherrill. He also includes both Southwest Hunter and Hunter in the range.