I have a slightly different point of view... the Bosh CIS fuel injection system is remarkably simple once you understand how it works. I agree that the need for enrichment during start and even worse during cold start might at first glance appear mysterious but it too is remarkable simple. And lets not forget... Mercedes did make the Bosh CIS fuel injection system... Bosh did... and it was used on many more cars than just Mercedes.
You can eliminate many areas of consideration by performing a simple fuel pressure test. If the system pressure and control pressure are within spec you can eliminate the entire fuel pressure side of the equation.
You can plug the cold start valve tubing leading to the FD and eliminate that possibility... especially in the warmer weather, the cold start valve has no function.
The early 380 M116 engines used K-Jet the later KE-Jet... either way, if the injectors are original... replace them, don't even bother with trying to test them, there are no components to check, they don't open and close like electronic fuel injection... they spray all the time as long as there is fuel pressure high enough to open them. The other part of their job is to atomize the fuel as it sprays. Injectors can get clogged or they can leak or they can not open due to insufficient pressure but little else can happen to them. Just replace them... it's a cost yes, but at 30+ years old... it's time.
That leaves the FD and on the K-Jet, the ridiculous after thought called the frequency valve. The frequency valve is a fuel injector but unlike the other fuel injectors the frequency valve is turned on and off electronically. You can listen to it working by touching a stethoscope to it. You will hear a very noticeable buzzing indicating that it is receiving the on/off pulses. If we assume that the valve is not clogged or not leaking then the frequency valve is performing it's function. You could disconnect the frequency valve from it's fuel source and plug the fuel source (figure a way with fittings) and see if it makes any difference in your running rich problem. If it doesn't then you can eliminate it as a source for your problem. The frequency valves only function is to adjust the mixture in order to pass emissions tests in the US and is unlikely to be causing a rich fuel mixture condition to the extent that you have described.
That leaves the FD itself and that will require a more complex set of troubleshooting steps. We'll tackle that if we get there.
Good luck