Same thing has happened to me a couple of times. I have found that if I get out a different car and track, run it for awhile and then go back to the original car and track that the brakes were locking up on most of the time the problem is solved. Hasn't happened for a while now but try it and let me know if it works for you.
I've driven that car, fully upgraded it but I didn't do any tweeks to it yet. I took it out on Monza and I was a bit surprised how much braking you have to do before you get to a corner because of the weight, once I got used to that, that car is a blast to drive. No problems here.
The day of the 2.08 update was very odd. Things that nobody ever experienced before, especially regarding tires and brakes, were going on that just disappeared shortly after.
So, reboot the console and test if your same issues still exist.
If you still have problems, then read through the Help post linked in my signature for ideas about how to deal with them.
After my last post, I have talked to several people that have the same car, and it appears to be a common problem. I even attempt to reinstall the game, and all DLC including GT-R the problem remains. Looks like a developer of this car needs to get some sleep and release a hot fix for it.
I took the car out at Trial Mt and found it just about undriveable. I was surprised by the weight, and it took a lot of tweaking to get good. If those four drivers were able to win their class after 24 hrs with the car handling like that, my hats off to them.
I believe I know why some can drive this car and others not. For some reason it thinks it a FR not a 4WD. When on acc only the rears spin. Watch a reply of your car coming out of the sand. And in the settings there is no center or front diff settings. I.e. , someone who drives FR more then 4wd would be fine, yet a 4wd diver would be lost as none of there tunes would work.
In your Garage when you pull up the profile of the N24 '12 it says under "drivetrain" - 4WD. Feels like 4WD to me. Car's just VERY HEAVY so you have to brake a lot earlier than usual. Took it in stock trim no tuning to Nurburgring 24H and did 9:07.xxx on Racing Hards... the tire it comes equiped with. That time was done with ABS set to 1... all other aids off.
In comparison I used a stock Shulze N24 '11 and did 8:53.xxx on the tires it's equiped with - sports soft. Was an enjoyable experience to run both cars in stock trim.