I am not fully sure if I fully understood the issue but it seems that you are not giving it enough throttle when the clutch starts to bite? I agree that in real life at least the cars I have driven most of the time have a more forgiving clutch. It feels as if LFS engines have very low inertia when it comes to starting to drive.
When I want to start driving in LFS with reasonable real life revs (at or below 2kRPM), at first I give very little throttle to raise the revs and when the clutch starts to bite, I give it quite a bit more throttle, like 40% while at the same time releasing the clutch more aggresively.
You can try simulating the issue in single player and saving a replay for us. To save a replay, simply press "2".