Newbie

Has she been stood for a while?

M102 seem to be prone to sticking exhaust vales when stood for a good while

The 4 pot engine can't cope with the lack of compression 

Often shows up with a puff of grey smoke that looks like steam even if it starts and runs lumpy will smoke and smell terribly 

Simple compression test should show it up 

So important I posted it twice ;0)

Doubt it. Pump runs, gets to pressure and stops. That's normal.  It will/should run again, and it is new so pressure should be sufficient (that's important).  This sounds like kJetronic control issue to me.  I had starting issues (ran fine once started) and had to take it to a KJet knowledgable mechanic in Reading (mainly just because they had the emmissions gear).  At the moment mine runs well but very rich and excessive CO, so I need to revisit it and probably change the injectors next.

The 230/280 KJetronic is mostly all self managing.  There are no real adjustments you can make manually, but there is one really sensitive one on the top of the fuel distributor which is the IDLE mixture which adjusts the position of the air intake plate to get the idle mixture right.  The system takes care of mixture when throttle is opened.

There is also a Warm up Regulator, (WUR), and other bits...Control Pressure Regular

Take a look at my saga here:  http://www.gwoa.co.uk/content/230ge-engine-stutter-misfire-jetronic 

because there are some good tuning reference guides there, like this one:

https://tasteslikepetrol.net/2020/02/bosch-k-jetronic-basics/

It explains what and how to test.

Have you checked the fuel tank itself, they rust from the top downwards and can fill with crud, it could have been that that killed your pump in the first instance. Quite straightforward to take off but I'd replace the whole tank with a later one from a 463, but it means you need to re route the exhaust too

AFAIK, there is nothing controlling that flap other than the downward suction of the pistons on intake.  It is lightly balanced.  If it is shutting off rapidly, what is doing that? Backpressure?  The idle adjustment allen screw (3mm) only adjusts the resting position (idle) of that plate, which should be approx 3mm below closed.  The kJet control module manages everything else.... I think it takes sensor readings to adjust the fuel metering which is controlled by pressure (control pressure regulator).  This is different when the engine is cold, or warm.   Its all explained in the Part3 of that 'tasteslikepetrol' link above.

I extract the following:

If you rev the engine and let the RPM drop, it should drop straight to a nice idle. If it drops and rises back up again, or hunts at idle, the mixture is wrong; this is the system trying to compensate. You also need to get the idle speed right at this point too, as altering the idle speed will alter the mixture demand.

By slightly raising or lowering the sensor plate you can tell if it needs to be richer or leaner based on how the engine responds. A small turn of this screw, with a 3mm hex key, will do a big change. Left is Lean, Right is Rich, so turn the hex key counterclockwise to lean the mixture out. With a little playing, you should be able to note the point where between rich and lean the engine runs best. The idle speed will usually come up slightly, and it will run smooth. Tweak the idle speed down if needed and adjust again until its happy. Once you have it in the happy place, you should now note that if you bring the revs up and let them drop to idle it should settle straight into a nice idle.

But I think you need to solve the fuel delivery issue.  Even getting a 12volt supply to it so it is running continuous.

Also, locate and replace the Pump Control Relay (also called the electronic control relay)... seems to be a common failure thing and might solve that pump problem.  

As Morecombe and Wise might say - it seems you are playing all the right notes, just maybe not in the right order!

At the limits of my suggestions.  But you have the new relay in.  And the pump still doesn't kick back in. So the assumption is fuel starvation causing engine to stop.   The idle control mixture (allen key) can be discounted because you get it running and once applying some throttle the idle mix doesn't matter.  Distributor and timing must be OK because it runs. So maybe its the ECU box that takes sensor inputs and presumably drives the Pump Relay and hence pump??

Otherwise.... Refering to the 'tasteslikepetrol' guide:

If all of that is working as it should, the car should run and respond fairly well. The biggest problem is when the fuel pressures are wrong. It only takes one wrong figure for the whole lot to fall over, and then its a case of tracking back and finding out where that pressure has gone. This is why people dislike KJet.

And the rest of the guide goes on to test pressure (cold, warm etc).  If that needs too much kit and hassle, I still think hard wire the pump to establish that with continuous fueling the engine won't stall.

Two other sources: This youtube guy is a guru and does a lot of videos about kJetronic.   https://www.youtube.com/user/schilleryu  (his english is painfully slow and a bit odd, but he is very good)

(Note there is a KE Jet and a KJet.  KE means Electronic module is present (ECU control).... I am assuming you have this type, as is mine '87 and '89 230GE )

THE KEJet ECU and Over Voltage Protection Relay is described here: https://youtu.be/uU1j_wcvLD0  (worth a listen)

This one is fuel pump relay: https://youtu.be/i45KMD13f2U  (jump to 9:30 to hard wire the pump without the relay)

This one is the Ignition Control Module: https://youtu.be/1UQjX83S8iA  (starts in German, but is English)

This is a general KEJetronic overview: https://youtu.be/4XeDlEjx5pc  (starts about 4:15)

Finally, I am not sure where the ECU box is, in the G, but if its broken...