I would think LSU is the pick in all your categories. However, no team in the SEC wins its division every year. If Arkansas wins the West this year, then there will be 6 teams that have won all the divisions since 2000, with none of them winning less than 2. I would have to dig further, but I think Auburn has lost more divisional tie-breakers than any other team. However, they usually lose a big game in November to drop them into the tie.
SEC WEST
National Championships - LSU ('03 co-champion)
SEC Championships - LSU ('01, '03), Auburn ('04)
Division Championships - LSU ('01,'03,'05); Auburn ('00, '04), Arkansas ('02)
SEC EAST
National Championships - none
SEC Championships - Georgia ('02,'05), Florida ('00)
Division Championships - Georgia ('02,'03*,'05), Tennessee ('01,'04), Florida ('00, '06)
* 3-way tie-break over Florida and Tennessee