I came on here looking for an answer to the exact same question as the OP, so I’m pleased to have found this thread. I feared that I would have to save up for a LMP car, but it’s interesting to read that the Corvette ZR1 LM Race Car and GT-R R35 Touring Car are viable alternatives (and considerably cheaper options too!!)
I’d dismissed the Corvette, as I have the Corvette Z06 LM Race Car, and it handles dreadfully, which was surprising as the Corvette’s were so nice to drive in GT5. I’d also dismissed the GT-R R35 also, as when I checked the line-up of cars for the GT World Championship, that car was at the back of the grid, with the front full of LMP cars.
However, thanks to the links above, I’ve found a tune for my Z06 which I will try later to see if that improves matters, although I suspect it may still be underpowered compared to the ZR1. Failing that I guess I’ll try buying either a ZR1 or GT-R R35 and see how I fare with those.
EDIT: The tune for my Z06 definitely made it a bit more stable, but I still struggled with it a bit, so I decided to go for the GT-R R35. I absolutely sailed through the championship and have been using it in the first few races of the Super series (or at least the ones where race cars are eligible). It's a superb car, that is both pretty quick in a straight line and super stable in corners. Highly recommended.