Can you provide a link to a site showing the specifications for that hands free kit?

I'm wondering what all the connections look like, whether it's powered by 12 volts or passive, whether the speaker is amplified by its own power or driven by the kit, etc.

It's sounding like some kind of a voltage problem where the handsfree kit is spiking the player and it's somehow resetting itself or there's some kind of a protection circuit.

The fact that it works if you start the call then plug it in later is very strange though.