A part of a solution: since pretty much any commercial mp3 player will
handle stereo, assuming sufficient channel separation, put your desired
voice on one channel and a tone on the other channel. Then use a
simple circuit to sense the tone and key a relay (or maybe even just
turn on a FET) to turn on the transmitter. Use a fairly high frequency
for the tone, and the response could be very quick. In simplest form,
the tone detector might not need any frequency filtering at all--just
detect presence vs absence of a signal. Difficulty is that you need a
way to put the tone on with the voice--easy enough if you can
pre-record all you need, but a bit more difficult if you want to change
the messages real-time. An advantage over simple VOX is better control
of the time the xmt line is activated.

I'm not sure if it would be any value, but you could also use different
tones to key different things; standard DTMF would be one way to do it,
but even just properly chosen single frequencies could work. And if
the player can reproduce high audio frequencies OK and you only need
voice bandwidth for your audio out, you could even put both the tones
and the voice on one channel, and filter the tones (or let the xcvr's
audio path-if filters do it).

Cheers,
Tom