I'd just get the stiffer front springs then and try that first. Front swaybars tend to be hard to remove/install(sub frame drop or removal), and stiffer rear sway bars tend to adjust the handling of the car(towards oversteer) more than they reduce sway on a nose heavy car.
I agree if you are looking to improve your time in autocross, but I think the OP is just looking for a car that has less body roll on the cheap, regardless of its effect on the balance of the car.But it's the nose heavy cars that suffer the most understeer and stand to gain most from a stiffer rear suspension. Doing so keeps the weight across the front more equal and gives the best chance of having grip coming out of a corner to get on the gas.
Yep if you've got stiff enough springs then some cars on some tracks do better without sway bars. Also a 90's Civic has double a-arm-ish suspension back there so it can dial in some negative camber with suspension compression.Posting an image doesn't help without pics of the suspension. There are roll center kits that can alleviate the roll center issues that lowered cars inherently have. It's a good fix for somebody who doesn't want to go through the R&D, manufacturing, time, and money for a shop to redesign the suspension.
There are other things that production based track cars owners do that generally go against the norm such as 90s Civics running without a rear sway bar (and actually getting better times.) To each their own.