Am I reading correctly that you have removed everything, including the LED OCD? If not, I would definitely suggest that first.

Disconnecting all the lamps and using the "testing with a jumper and bulb" section of that pinballrehab page is a good way to narrow it down between a board or wiring problem. Even better than a jumper is this test board:
http://www.siegecraft.us/presta/index.php?id_product=11&controller=product&id_lang=1

Often when installing LEDs, you can bump a socket and create a short. Occasionally, you can also get an LED bulb that has a short internally. That has bitten me twice now.

It sounds like you have a couple of things going on.

First, I believe you do have a problem with the game wiring that is related to the start button. Disconnecting it should not affect the other lamps. Since it does, that leans toward some sort of problem. I would remove the LED OCD for now. It is an added layer of confusion when trying to troubleshoot problems. Without the LED OCD, you said it works fine with the start button disconnected. If you connect it, it causes problems? If so, I'd solve that first.

Second, it sounds like one or more transistors on the LED OCD have been killed. If manual test is causing double lights, you can be nearly certain that the "extra" light has a transistor that has failed short. Say you turn on lamp 13 and lamp 83 turns on with it. That almost certainly means that the column 8 transistor is failed short. I say almost because you still can't really trust your game wiring. That's where the test board comes in. You can trust it, and it displays things in an easy to understand pattern.

Once you figure out which transistors are dead, you have to have someone who can do surface mount rework to replace them. They are inexpensive parts, but they are very small.

I would have to think that finding someone in the UK to do it would be much cheaper than shipping it overseas twice. The following part are what are installed by the factory, but nearly any MOSFETs with similar specs would work.

Columns, Q1-Q8, Vishay SI2319CDS-T1-GE3
Rows Q9-Q16, Infineon IRLML6344