I wouldn't recommend Mineral King for any trailers even small. Some years ago my Grandparents were hosting there (for about 10 years) and they literally closed the road down for them to drive up their fairly small Class A motorhome.. and even with not traffic it was a challenge.
Last time I drove through (last year) with a full size reg bed pickup... my wife was not pleased... we had some close calls with idiots cutting corners as well.. We'll never be back there.
Probably a bit late now, but in case anyone stumbles onto this thread looking for similar info, if coming in from the West forest route 14S32 off generals highway 36.692180, -118.870143 has a several dispersed camping sites..some very nice. No water or services of course.