In regards to the stutter this might help,
This was on another forum, sourced from a manual, dealing with a different bmw with a afm. They were not going to maf, just saying its the same basic setup and someone was having a stutter about like yours. And here was one of the possible causes..
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"The shut down happens if the air volume sensor registers an air volume that is higher than permitted for current RPM and load, for more than 1.5 seconds. The limit is at 15% above the volume at full throttle. When injection is turned off, the compression process collapses immediately. Injection is resumed when the air volume sensor registers a pre-determined value that is below the full throttle amount. This new value is maintained for 4 minutes as a cut-off value forthe injection unit.Engine performance is slightly reduced during this time. After 4 minutes,the original value is restored."
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So possibly, if this bmw works like the other, by 3k rpm, with the new maf, the cpu reads a higher air volume than is permitted for the RPM. This could mean your parameters are set too high in the 1k-3800k RPMs due to how much more free flowing the maf is (especially around idle), the (voltage?) reading might need to be lessened to look weaker so it doesnt trip this air/rpm ratio.
If there was a way to get the car to say 4500RPM before 1.5 second of turning the car on, in neutral, you could test this but 1.5 seconds is pretty short. If it holds the value for 4 min, probably wait 5 min before tries...
Ive been watching this thread for a while and really want to see it happen. Good luck with it, hope this helps..