Well as you asked for it, it doesn't matter! I've used Engine Management Modules from the 2.8 Lt Twin Cams on a 250 Single cam without a problem.
There is one thing to look at on the left hand side under the dash there should be a group of brown wires which go to earth make sure that these are actually earthing and have not corroded as that will give you exactly what you are describing.
For a run around I got a Pug in for repairs and was told that after he drove through some water it stopped. As the fuel pump was not running it seemed a good place to start so I changed that no improvement then it was suggested that a relay under the dash might be faulty and as there is no way of testing except to change it a new expensive relay was brought but again no difference. I even by this time had disconnected the fuel filter just to make sure that it wasn't blocked and building up pressure in the filter and stopping the fuel pump from running. Well after listening to the owners description of how it happened I gave up and started on the normal diagnostic things and found a bad earth for the fuel pump inside the boot.
Apparently the car had been running badly for quite some time {of course I wasn't told this} and when there is no solid earth provided for the fuel pump it sends out a brief pulse of power and if nothing happens the management module shuts down the fuel pump which explained why I was not showing any power present at the fuel pump.
Go over all the earth points pull them apart clean them up and then try again.
As for the Cold Start Injector even in Darwin it doesn't get hot enough for it not to run at least on startup so I would look very closely at the relay which controls it but only after cleaning every earth point you can find and making sure that it is actually earthing. I take it that the Ignition Module is working as you said you had it running on the Cold Start Injector but new points never hurt. And while you are at it check the cam chain as well. They are a pain but do quite often need changing and are not all that expensive provided you stay clear of Mercedes and buy from
http://www.mbspares.com.au/default.asp?d=18036&p=16933
In Canberra. They are quite helpful the last cam chain I brought from them cost me about $80.00 AU freighted to me overnight. They should also be able to help you with a engine control module as well but I would go over every earth point that you can find before taking that step. Most of the troubles in any EFI car relates to earthing problems and I've seen many a mechanic replace some very expensive items with no result only to find a bad earth point at some out of the way place.
Provided the Box looks the same and has the same connector it will work although the Auto Transmission may change at different points in the speed range depending on just how highly stressed the engine for that model that the module came out of is.
If you have to change any electronic parts remember to disconnect the battery first and grab some metal part of the car to bring yourself to the same potential as the chassis that way you will avoid Static Damage which is another killer of these electronic components.
Have you checked the points on the side of the distributer as if these are sticking or have lost their spring tension they will not be sending the signal to open the main injectors as well.
I hope that is of some help to you just post again if you need to know anything else.
Just as an after thought a lot of the Power Steering return hoses are replaced with water hose which reacts with the ATF and makes a real mess under the bonnet so if this is the case replace the return hose with the proper Oil Hose and clean down everything in the engine bay before the ATF begins attacking all of the rubber parts like the relay isolation washers which are very hard to get.
Col