I am no expert on Mineral King. It was my first time there. However, I don't think that you want to go _up_ Sawtooth Pass to get to the Big Five Lakes Basin. I mean, it is possible, but I wouldn't want to do it. The west side of Sawtooth Pass is an awful slope of loose scree and no maintained trail.
I started from Mineral King, then went over Timber Gap and Kaweah Gap, then around to the bottom of the Big Five Lakes Basin, then continued around to Lost Canyon, up to Columbine Lake. Sure, that is a lot of miles to get around, but it is better than knocking yourself out from Sawtooth Pass (the hard way).
I had originally intended to go over Timber Gap and Kaweah Gap, and come back west by way of Black Rock Pass. But then at my first camp, I was studying the topo map, and the descent off Black Rock Pass, then the subsequent ascent to Timber Gap again did not really warm my heart. So, I made a sudden decision when I was near the Big Arroyo Cabin, and I went farther around the way I did.
In fact, the whole trip was a spur-of-the-moment thing. I had a different trip planned with somebody else who suddenly bailed. I had the gear ready to pack up, so I left within the day.