As someone who lives in Bear Valley part time, I can tell you with 100% certainty that you are not allowed to drive past the gate or a closed sign. Even in the spring when the roads are partially clear, driving on them will get you a ticket. That goes for a lot of the forest service roads as well.
Also, I wouldn't be overly confident in your ability to drive on the groomed roads if you decided to proceed. Down in the village the roads aren't plowed; they're groomed for snowmobile, ski, and foot traffic. Every year someone with a 4X4 thinks they can make it and ignores the road closed signs. They never make it more than 30 feet before sinking to their axels. Multiple tickets (driving past the close sign and damaging the road) and a very hefty tow charge are their reward.
If you do want to give driving on snow a try, I recommend the road to Highland Lakes in early spring. It's not plowed and is generally snow-covered past the campground. Just be sure to bring a shovel and a friend with a winch.
Highway 4 is supposed to open sometime around April 4 this year, which is a month earlier than normal.