The charging circuit for the key battery is completely separate from the RFID transponder pill stored in the key.
This is why the valet keys work for ignition, yet they contain no battery or standard RF transmitter for lock functions. It's also why some people buy the valet or wallet keys for cheap, an ebay key for cheap, and install the coded RFID pill into the ebay key. That's a lot of work though, and it ends up being about the same cost as just buying it from a dealer because the key blank must still be cut by a locksmith. I paid $120 at BMW of Peabody, MA with my CCA discount a few months ago.
My last car, an e46 2000 328i had a diamond key too which started to have range trouble until it finally quit. As a stop-gap measure before I installed a new rechargeable battery I would put the keychain on top of my toothbrush charger at night. Kind of weird, but it worked. The key does charge by inductive coupling but my guess is that the car stopped charging the key in the same way that many standard battery chargers will not recognize a completely flat battery. The OralB toothbrush charger though just emits a field regardless of whether something needs charging. This let me have a fully charged key all day that worked like new.
I believe the key remote frequency is 315Mhz, does your spectrum analyzer have sufficient bandwidth and an antenna to receive this frequency? To make things easier, most auto parts stores have car remote battery testers at the counter that will illuminate a light when they see a signal. Those work well to check our keys.