I think the Mineral Creek area is a longshot in late May-early June. We're having a pretty normal winter (above average snowfall so far), and there will likely be a lot of snow on the road that time of the year. For reference, the Ice Lakes Basin trailhead is 9850' elevation. The area your looking at is a little bit higher, probably closer to 10,000'.
We are currently in a "La Niña" weather pattern, which typically means that the big storms, especially spring storms, stay north. I would frequently check this website for info on the snowpack:
http://www.co.nrcs.usda.gov/snow/snow/watershed/current/daily/maps_graphs/swe_time.html Use the pull down menu for the San Juan, Animas, Dolores, San Miguel basin. We are currently at 130% of average snowfall. Check it in February, March and April. If it dips below average, that bodes well for accessing the Mineral Creek area. If it stays above average, I would suggest looking for a lower elevation basecamp. Check out the West Fork of the Dolores River. There's a good road that runs up it, it's lower altitude, and the river is very nice. There's good camping, hiking, and some offroad driving.
Mostly, you need to keep an eye on the snowpack. Access to the San Juan Mtns high country that early is pretty iffy. Good luck!