The Formula BMW has an automatic clutch. Just lift the throttle and shift, don't use the clutch on your wheel.
Turn 2 complex needs to be 3rd gear, not 2nd. The gearing everywhere else seemed fine.
You're braking too hard and locking the wheels. When you hear the engine get quiet, you're braking too much. Wheels should be making a "chirp chirp chirp" sound, not "screee".
Trail-braking is way too aggressive - you're parking it in the middle of the corner. Don't trail brake to begin with, just focus on carrying speed through the corners. Once you have the speed, bring your braking point down and fine tune your brake release as you enter the corner (trail braking).
Find the speed through the corner. As I said you're parking it at the apex. Get to know "slip angle", Ive done a bit on general cornering on
my blog. It's more about driving real cars, but 95% of it is applicable to LFS.
You're also apexing too early. The only thing you do with the wheel after the apex should be unwinding. Don't hold or add steering lock. If that happens, that means either your angle at the apex is too little, or your apex itself is early.
The good news is, you never got loose (oversteer) in that lap. That's good, and it shows a trait of sensetivity on your part.