Go for it. I have AutoX's for years with top down and love it. When I was Road Racing AI we had a guy in CMC with Foxbody Vert, a course full rollcage, but I always love his car and he did well. He was from Texas and moved to S.Ca.
Bar not required for AX. I have a friend with SN95 Cobra Vert and he does well with top down, said it balances the rear weight and helps the handling. If you don't have passengers, put a rollbar in and do both, AX and Open Trk, but with your kids do what GTRacer said.
1993 Fox C4, carb-347 Pro-brkt car M10's
700+ elim rnds