Presuming you intend true winter mountaineering conditions...
I'm not familiar with Dinkey Lakes. Rae Lakes loop could be a great winter trip. You will need to go from the East side since the road to Kings Canyon won't be open. The road to Onion Valley won't be open either, but you can drive as high as possible, drop your gear, then descend to the road closed sign at about 6500'. (just in case a typical March storm blows in and lowers the snow level)
Kearsarge pass is fairly easy in winter. Glen is another matter. I suggest using Kearsarge to access Bubb's Creek and Dragon for Rae Lakes.
Snow line is unpredictable but right now on the Owens Valley side it's about 7000' on north facing slopes and 9000' on south facing. Last week it was 5000'!
Much of your hike on lower Bubb's and Woods creek may be free of significant snow. With the low snowfall this year and cool March weather I think the creek crossings should be manageable. Watch out for a warm spell or worse warm rain. Higher elevations will have several feet in many areas. If no more storms hit you might get by without snowshoes but I wouldn't risk it. It hopefully goes without saying that ice axe, crampons, gear for below zero F, and appropriate skills are needed.
Last year a late March storm dropped several feet of snow in the Rae Lakes. Be prepared.