I used to auto-x a '87 GXL. The car has way more grip than the Miata or 2nd. gen. MR2, but it is heavier (N/A 86-88 coupes [a.k.a. Series 4] were in the 2600-2700lb range depending on options, N/A 89-91 coupes [a.k.a. Series 5] were in the 2800lb range). The LSD helped it power out of corners but the lack of low-end torque meant I had to keep the engine screaming or else it would bog down easily. The power steering was slow and the steering wheel is huge- a 360mm steering wheel helped a little. Manual steering racks are quicker -I have one on my turbo project and it turns in more quickly. Disconnecting the rear sway bar helped it turn in a little. Koni or KYB adjustable shocks are the way to go. Brakes are excellent, esp. with stainless steel brake lines.
I am a novice, but even an experienced autocrosser had a hard time in my FC against the Miatas and MR2s. A later 89-91 model might fare a little better since they had better low-end torque, faster steering and a better power/weight ratio in spite of the extra bulk, but don't expect any miracles. You might be competitive in a small club. I raced my RX-7 in a very competitive club (New England Region SCCA) and I was little more than an also-ran.
There was a rare GTUs model that competed in STS at this year's Solo Nationals but the best it could muster was 14th place. This was the driver's first year with it, so that may factor into the placing.
I think a well set-up FC could be a dark-horse in STR. The more open rules really help the car since it is so choked up stock. FCs can take a lot of tire with only a little fender rolling and they respond well to ECU tuners like the R-Tek.