If you really want to get all four, I think Mark's suggestion about Burnham Hollow is a good one. If you have a second car, I'd leave one in McKinley Hollow and go around and start from Burnham Hollow. About a month ago, T-max and Hike the 115 got three of those four peaks using a variation of this. They came in from Burnham Hollow, bagged Fir, Big Indian and met me on Eagle. I had 'whacked up from McKinley Hollow and we returned to my car there, where I shuttled them back to Burnham.
If you come over Big Indian to Fir and then have to reclimb Big Indian, it could be tiring. Climbing the Big I from the Fir side goes on for a while-it's flat, steep/ledgy, flat, steep/ledgy, flat, steep/ledgy...
As Mark points, if you start from Burnham, you get the orienteering and 'whacking in early, and walk out on trail. You also avoid the McKinley Hollow brook crossings until the end of the hike, when it's not as big a deal if you get a little wet. Not that I would know anything about getting wet on a brook crossing. Ahem...
One drawback to using the Burnham to McKinley route-it leaves Balsam for last, and I know you want that one because of it's "winter requirement" status. Which brings it back to the whole issue of doing all four in one day.
Matt