Ive been up there 5 times in winter, 4 of them solo. I would not recommend going solo. You cannot do all three peaks in one shot, even if you do camp at Bradley Pond lean-to. The only exception would be IF the trails are well broken and you don't mind hiking by moonlight. These 3 peaks are barely doable in summer with long long daylight.
You MUST pick a time when the weather is clear, not too windy or cold.
For snow typical in January, you will not be able to follow the herdpaths unless they have been broken and rebroken after each major snowfall. Otherwise you're on your own compass and map skills. You CAN follow Panther Brook about half-way up, but afterwards it peters out and things get confusing. If that is what happens to you, stick to woods on the left (south) side of the stream where there are more open, tall evergreens.
On one solo attempt I made it to Times Square with a full pack but a storm came in and it was whiteout everywhere. Temps dropped to -20. (It was legal to camp there at the time.) I headed back the next morning and was barely able to follow by previous day's path. On the way back, after leaving the brook, I got lost under the cliffs and spent almost 2 hours trying to figure out my way back. In other words, it can be dangerous up there.
I was finally able to do all 3 peaks in individual winter as dayhikes, but I had to walk up to and back from Bradley Leanto in the dark.
You could, I think, potentially do all three in one long trip as follows:
Start your hike in the dark so that you make Bradley Pond leanto by first light. Work your way to Panther Brook and then up to the legal 3500' level. Make camp there. This of course assumes you can find your way to that point, and the snow is deep enough to stomp out a flat campsite. Then hike up to Times Square and do Panther. Don't linger. Then do Santanoni. Then descend back to your campsite. Next day go back up following your previous trail to Times Square and do Couchie.
I wouldn't try any of this though unless there was 100% visibility. The herdpath to Santanoni will be filled in with snow and unrecognizeable. But you can see the peak and just head to it. It would be hard to get lost on the trail to Couchie because it's on a ridge that is steep on both sides. But there are small cliffs that could be difficult to negotiate. And the section from the point (the vly) upwards is tricky until the ridge narrows again.
No special gear is needed, just snowshoes and poles. I doubt if crampons would even be needed uness you hike the brook which may have exposed ice or if the small rocky scrambles to Panther are icy. More likely they'd be snow-filled. Although I had them, I never used my crampons. Most needed is good weather.
Keep watching the postings here and you may luck out in following a big group a day or two ahead of your planned trip.
oldfogie 3102W