There's no doubt that this challenge is properly hard.
I second the suggestions to take a break when frustration sets in. The point where tiredness overtakes the learning process is where I try to have a break. (Trouble is, this point is usually long gone before I realise what's happened, and I find myself in a groundhog loop, crashing at the first corner again and again.)
Practise is the obvious answer, but it is the right one. I'd say to start deliberately slow, and work your way up to faster times. That way you can guarantee yourself some clean two-lap runs, and build up more of a rhythm than if you're falling off the track at random intervals.
I didin't find I needed SRF for the bronzes, but I've been trying it out in my attempts to go faster, and it does seem to have helped get a silver on GP and a gold on Monza (cut chicanes, I'm not that good). It seemed to make it easier to brake deep into the corners, and certainly helps keep the car from skipping off on some of the bumpy sections of track, and even some of the curbs.
I use a wheel, and I imagine it's better for those more delicate sections, through the chicanes, where you need to point the car accurately before nailing it. I'd suggest choosing one control method and sticking with it, though. Definitely don't switch between the DS3 and the wheel during the same gaming session.
I'm not sure, but there's a slim chance that for some people the challenge is so fast that the lag in your TV/display might have an affect. If your TV has any "smooth motion" type effects, you could turn them off to see if it makes any difference. I'm speculating here, so I wouldn't be surprised if I'm talking rubbish.