I had that problem on my E46 Touring (same handle/switch design), and after removing the rubber part where you press with your fingers, I found what was wrong.
Under the rubber, there is a plastic piece (kind of 'L' shape) that pivots on a pivot point and the other end pushes on the micro switch. That plastic part was unsnaped from the pivot point. I reinstalled it and it worked since then.
For the non functionning licence lights, open them. Big chances that you will find rust, dirt and corrosion in there. Clean, sand, lube with some grease, install new bulbs, and it should be ok. If you have a voltmeter, test to make sure you have voltage.

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