I won this first try. I did have Skid Recovery on, though. Was able to lap consistently around 60 seconds, best lap was about a 59.4.
I used the '02 RX-7 Spirit R Type A (FD) from the new car dealership.
Mods:
All weight reduction: Stage 3, window, hood
Chassis Reinforcement
ECU, Stage 3 engine tune
Sports Intake Manifold
Sports Exhaust Manifold
Sports Catalytic Converter
Sports Exhaust
Custom Transmission
Twin-plate clutch
Semi-racing flywheel
Adjustable LSD
Carbon driveshaft
Custom Suspension
Settings:
Aero : 0 (didn't realize the car came with aero kit, so didn't touch it)
Transmission: top speed 240 km/h (about 149 MPH?)
LSD : 40/60/5
Brakes : default (5/5)
Suspension settings:
Ride Height : -10 / -10
Spring Rate : 12.6 / 9.5
Dampers (E) : 5 / 6
Dampers (C) : 5 / 6
Anti-Roll Bars : 3 / 4
Camber : 2.5 / 1.5
Toe : 0.00 / 0.10
EDIT: Without Skid Recovery, and with these settings, I was able to get a 1:01.7 after a few laps of practice, and I think I could get into the low 1:01s with a little more practice. Which probably wouldn't be enough to win this race... If somebody can win it without Skid Recovery on, my hat is off to you...

That matches up well with what JacobN posted a few posts ago, that Skid Recovery is probably worth about 2 sec per lap.