BMW uses, believe it or not , mitsubishi starters and my r55 is equippec with one. Per your post it seems that the solenoid is clicking but do you feel the engine trying to be turned by the starter? If yes , go under the car and it should be onthe upper right corner of the tranny and spray it with wd40 then try again. If it is still not working get a hammer ask someone to crank while your banging on it, im not talking about tapping or demolishing but hammering with force, this works when the armature is coated with gunk , oil and dust so it sticks.id say 40 % of the starter problems i get here at the shop are solved with the hammer. If nothing while asking your helper to switch the ignition to start position feel the positive cable if it gets hot immediately , replace starte
I'm guessing you are correct. Any time you have a starter problem always check that all connections are good and tight. This has bit me more than once. My daughters car worked for a long time, then she told me it wouldn't start. It was a lose battery cable. I'm glad we could help.