I think 1 1/4 tubular would be fine. You don’t have the really stiff t-bars.
Do you have a rear bar? An adjustable can help balence it out.
I have much larger 1.14” Tbars and 1 1/4” tubular front sway bar. I have a rear adjustable sway bar. But at Willow Spring Raceway the rear had too much roll stiffness that gave an oversteer/loose condition.
So I disconnected the rear bar all together. Much easier to drive and no sliding into the infield.
Now for a autocross with lower speeds and tighter corners, the rear bar may help.
Right now, I drive it with the rear bar still disconnected on the street.