Whitney in winter can be done... if the snow isn't so deep that you are swimming uphill in drifts. People go in January, but the conditions really make the difference. If it is a dry year, you can walk through the snow that isn't too deep. In a normal to heavy snow year, you'll be swimming and make maybe a mile in a day. That's when a group of strong mountaineers is needed so you can keep trading off the lead.

I saw a recent mention of the mountaineers route, below the Ebersbacher Ledges having chest-deep snow. No surprise, as it drifts off the cliffs above and piles at the base, right where the MR route proceeds.

Several guide services take people later on, when there's enough snow in the route to bury the brush (they skip the Ledges), and they can pack a path over the top. But the key is waiting for significant snow, and then waiting for it to consolidate so you can progress over the top of it. They use ropes to both climb and descend the Final 400.

If you're looking at this month or next, it might be better to consider the Main Trail route, and get as far as you can. But beware that there are avalanche zones there, and on the MR. I've seen reports in the years past of several groups barely surviving an avalanche. (One guy survived an avalanche on the Main Trail, then rolled his car on black ice on the way down to Lone Pine. Another group camped in the valley below Iceberg Lake, and were nearly buried while they slept in their tent!)

You won't find much beta on anything on Whitney now, since most people just stay away.

And finally, written long ago by Bob R, who climbed Whitney many times, and was a member of the China Lake Mountain Rescue Group:
          So, you want to climb the Mountaineer's Route in winter?