Many mechanics are afraid of these carbs because they don't understand them. Mixture richening for acceleration is performed by electronically closing the choke flap, rather than squirting an extra dose of fuel in!
From personal experience, I would recommend repairing the Pierburg carb.
I had an '87 model 316 and made the big mistake of getting the Weber 34/34DMTL conversion.
I hated it. The throttle opening on the Weber is linear, whereas the Pierburg is exponential. What this means is that you can redline the engine with about 1/8in of pedal travel. Not good for city traffic jams.
Oh, and economy went down from average 33mpg to 28mpg.
The Haynes Solex/Pierburg Carburettor manual (UK) gives detailed setup instructions.
If the screw heads on the adjusters are snapped off, these are available at BMW dealers (special order).
Maybe there is a small vacuum leak. Thoroughly check ALL pipes and hoses, refer to the manual to see if they are in the right place.
Check the DECELERATION AIR VALVE. This is the thing that connects between the manifold and a stub on the air filter pan. It has a third hose going to the base of the carb. There shoud be NO vacuum on the air filter stub at idle. It is needed because unlike a conventional carb, the throttle butterflies are COMPLETELY shut on deceleration, the valve then admits fresh air into the intake to avoid over-vacuum.
Also check the filters in the throttle actuator and the outgoing connection to dash temp sensor.
Check for vacuum leakage at the throttle butterfly pivots. If applying some oil to these restores smooth idle, you need to replace the throttle plate. However that part will cost you the same as the complete Weber conversion kit.
A small cheat tip:
If you just need to raise the idle speed slightly, there are two single wire connectors taped back to the harness adjacent to the ECU under the dash (to the right of the steering wheel, UK RHD model). These can plug together and raise the idle speed as the ECU now thinks you have switched the Aircon on!
Hope this helps,
Si