It's not the tires that cause your TC to attract the DUI enforcement guys, it's the steering. If there is ANY appreciable wear in the suspension components that will only compound the problem.
Experienced TC drivers will tell you that you don't really steer a TC. You suggest where you'd like to go, and the car interprets that in a rather free form manner. It's rather more like sailing and less like driving.
There are several upgrades which improve the system dramatically, but which the Originality Police won't like. One replaces the steering box with a Volkswagen one. Another uses a Datsun box. The most "acceptable" improvement is the Tompkins Steering Kit. It replaces a couple of the components of the original steering box, but the results are not as dramatic.
Let me stress again that if you want this car to go down the road straight (straight is a relative term here) EVERYTHING in the front end, from the spring bushings to the wheel bearing adjustment, must be optimized. If the kingpins are at all loose you'll have a handful. If the tie rod ends are at all sloppy it'll steer like a dump truck. If everything is PERFECT, a stock TC with the Tompkins kit will drive pretty OK.
Do take Dick's advice and talk to Ben Cordsen. He's a great guy and he'll have a pile of tips and information for you, including info on joining a group online just for TA-TB-TC cars.
Motorbill
From Lola to Land Rover, If it's British and has wheels, it's likely I've bloodied me knuckles thereupon