Do you have R.J. Secor's book? "The High Sierra: Peaks, Passes and Trails." This is the authoritative book on x-country passes in the Sierra.
In case you can't get it in time, I'll read you some info on the passes in question.
1. Glacier Pass - yes, this is off-trail. From Secor: "11,080+ ft. Class 3. This route connects the Sawtooth Pass Trail with the Black Rock Pass Trail via Spring Lake...From Groundhog Meadow along the Sawtooth Pass Trail (8757 ft) leave the trail and hike directly up the slope above. (Alternatively, you can remain on the Sawtooth Pass Trail beyond Monarch Lake.) You eventually regain the Sawtooth Pass Trail below Sawtooth Pass. At the point where the trail switchbacks to Sawtooth Pass [note: it's more like a confusing smattering of 10 different use-trails], leave the trail and head for the saddle to the north. The north side of Glacier Pass is a short, steep cliff, which usually has a small snowfield until late in the summer; in the spring there may be a cornice blocking the way. Descend on the north side of the pass and go down to Spring Lake, circling it on its north side. Remain above the 10,000 foot contour and make a level traverse to the Black Rock Pass Trail."
"When approaching Glacier Pass from the north, follow hte north side of the stream that feeds Spring Lake to the tarns in the meadows below the pass. During low snow years it may be possible to skirt the snow on the right (west) side of the pass by climbing steep talus."
I hope someone who has done this pass can chime in. Having eyeballed it from both the north & the south sides, it looked very easy navigation-wise (everything is line-of-sight), but that snowfield on the north side looked freakin' steep and exposed, and still substantial in August last year.
2. Cyclamen Lake Pass - this is the pass that goes west from the Big Five Lakes basin, and then down to either Spring Lake or Columbine Lake. Secor's description starts at Columbine Lake. "11,145 ft. Class 3. This pass has also been called 'Hands and Knees Pass' and 'Bunny Ears Pass.' This difficult cross-country route leads from Columbine Lake to Big Five Lakes. (It is the author's experience that it is easier and faster to hike down the Sawtooth Pass Trail through Lost Canyon and then take the Big Five Lakes Trail.) From Columbine Lake, cross the shallow pass to the north and descend steep slabs and talus to the east shore of Cyclamen Lake. Make an ascending traverse from the lake to Cyclamen Lake Pass; this involves more steep slabs and talus. From the top of the pass...descend to Big Five Lakes."
3. The third pass, already mentioned by someone above, is the unnamed pass between the Big Five and Little Five Lakes basin. That one (to the eyeball) looked easy and fun. I wish we had done that, instead of taking the somewhat boring official trail through the woods between the two lake basins.
You actually sound like a strong group of hikers, but if you aren't sure how far you can go on day 1 given the altitude and the ruggedness of the trails (even the official ones) in Mineral King, I stick by my recommendation to just go over Sawtooth Pass on day 1 and see how far you get. The climb to the top of Sawtooth Peak is a must-do side trip, for many. Columbine Lake is a stunning place to camp! Lost Canyon is gorgeous too, although there were fewer obvious camping areas there, and it's meadowy so there may be skeeters. But Columbine or Lost Canyon or even the lowermost Big Five Lake are all attractive and convenient camping bail-outs should you not make it all the way into the main Big Five Lakes Basin on day 1.
From Big Five Lakes, you will have some options. Personally, I say don't miss the uppermost lake in Little Five Lakes Basin, nor the descent down Black Rock Pass to the west. Gorgeous. From at or below/west of Black Rock Pass, you'll have a line of sight to that snowfield below Glacier Pass. If it looks do-able, you can go out that way. If not, you do the longer walk, on-trail, via Pinto Lake and Timber Gap, back to your car. (This trail walk is only partially shown on the map you linked; but it's well-marked and will be fast walking compared to your last few days).