Is It Possible?........

Cornering ability is really just balancing suspension stiffness with tire grip. If you're limited in tire selection for your setup, get the best tires that you like and then set up the suspension around that. Before I put 18" wheels with low-profile tires on my Duster I had 15" Rallyes with 60-series Firestone Firehawks (the older street-touring ones, not very grippy) and my 1.00" torsion bars seemed to be a good matchup. There are ways you can calculate what your spring and shock rates should be but it's probably easier to just trial-and-error it. TBH you won't really know how your car handles and what upgrades it needs until you push it past its limits and there isn't really a place on the street to do that legally or safely unless you can find a BIG open paved lot somewhere near you far away from people.

One thing I've found with my Duster, it's been really tough to get the rear stiff enough to not always want to understeer when it's pushed hard. I got Hotchkis leaves and they are still way too soft, have my rear sway bar on the stiffest setting and it's not nearly enough roll resistance. These cars REALLY do not like to oversteer at all the way they're set up from the factory. Not that you want oversteer but you want a balance where you can have a bit of both depending on the situation.