Still new to this...
I am having what I think is a similar problem. My outside temp reading has dimmed and looks to also have dimmed part of the odometer display (this is on an E39, instrument display).
Is the fix for this similar to what you have discussed?
I'm pretty good with the mechanical side of things, but electrical scare me a little.
Are you in Wellington at all? if not, is this something that most competant electricians could handle?