I don't know where in Va you are, but over half of the state doesn't have any rifle bear season. Archery yes the entire state, smokepole season in half the state, and rifle season in half of the state.
Better to be sure. Here are the regs.
http://www.dgif.virginia.gov/hunting/regulations/bear.asp
and I would agree that a .243 is a bit light for a good sized bear. If you hit him, be sure of the shot and make it a good one.
Mitch