The problem is in the transition from the cold start system to the
regular operating air fuel ratio. That intermediate step means running
the still cold motor in a somewhat enriched state. Adequate fuel
enrichment during warm up is missing. But let's first look elsewhere.

Has anyone checked the engine intake air warm up - after the air flow
sensor and before the throttle some air is diverted to the cylinder head
or exhaust manifold to be warmed up so the air fuel mixture can be
leaner than it would be with cold air. If this "stove" isn't working as
it should be the ECU, getting its  engine temp signal from another
sensor, will lean the mixture expecting the intake air to be warmed but
it may not be warm at all so the air fuel ratio will then be too lean.

There may be a temperature sensor at the air flow sensor which is
sending a false signal. (I don't KNOW if there is such a sensor on your
motor, but it's worth a look.)

You've certainly invested a lot of money in this problem, hope these
ideas help.

When cold (oxygen sensor not yet warm, the engine control is operating
in open loop mode, i.e. it will regulate the mixture according to
various sensor inputs but without feedback from the oxygen sensor.

The basic mixture is adjusted by a sealed (plastic seal) adjustment
screw, The screw is located next to the throttle body and is accessible
through the long tube ("tower") from the top of the air filter
enclosure. The screw is spring loaded and must be pushed down to be
adjusted. It is factory preset to provide close to correct mixture
without ECU control, and should normally never be adjusted afterwards.
In fact this is how the non-electronic injection worked... the ECU with
EHA (ElectroHydraulic Actuator is a later add-on from K-Jetronic to KE
Jetronic (or CIS to CIS-E as Mercedes calls it). The mechanically
adjusted mixture will then be fine adjusted by the EHA according to
input from the ECU.

In your case, the mixture is obviously not correct in open loop mode,
and this adjustment could be the suspect (maybe someone adjusted it?).

Another cause could be a vacuum leak, but that would also be the case
in close loop mode when engine is warm, and it appears as if you
already checked this.

A fault in the EHA could create similar symptoms (if it is somehow
sticks near neutral position).

And another thing I can think of is the potentiometer, which gives the
Air Mass Sensor position input to the ECU. But that would probably also
give even worse problems when warm.