Help please x1 rough idle

[annahint01]

My 64 plate x1 s drive auto has my bmw specialist mechanic baffled and I’m hoping someone else has experienced and resolved the issue I have.
It starts fine and drives fine until 3 miles approximately. Then when I stop at traffic lights, jams etc it sounds like hunting but revs don’t move. I feel a shaking/strong vibration through the whole car and can actually see the shaking from outside. It doesn’t happen every time I idle I would say it’s about 7 out of 10 idles. Could do it at 4 stop lights then not at the 5th then again at the 6th.
I’ve recently noticed smoke coming out of the front and can smell fumes inside but don’t believe this is related as my mechanic has found the issue causing this and ordered a part.
Once moving it drives well. The engine and gearbox have been checked. It’s just idling I have the issue.
If I turn the engine off it often then starts without the issue for a short drive.
No codes when checked.
Thank you

[maxwmiller]

I'd go see a new mechanic. Heavy vibrations could be either misfires, rough running, or motor/trans mounts and if he can't diagnose those, you need to find someone better.

Auto stop start is consistent for me, either not happening or happening due to battery power, outside temp, etc.

Smoke is likely fluids on the exhaust manifold, very common to have oil leaks out of engine cover gasket or broken engine cover.