I think the RCM is fine. All the clicky-clicky sounds you hear are the throttle body doing it's powerup dance, the fan in the RCM box will whine with the key on, and you MIGHT hear the 'vacuum cleaner sound" from the secondary air pump.

Starter failures are RARE with these cars. You can prove the starter by getting into the fuse box, under hood, driver's side.
Pry up the fuse carrier deck, lift it up a bit and grab the pulse module. Unplug it, and pry the covers off.
Reconnect the pulse module, with the cover off.
Verify car is in park or neutral and the brake is firmly set (you are about to risk the car running over you).
Turn key to "ON".
Now, there are two relays on the pulse module.
Press both, one is for the SRT coolant pump (if your car is not an SRT, you will have the relay but it won't do anything).
The other is the relay that directly drives the solenoid on top of the starter.

If pressing that relay makes the engine spin, you have a good battery, starter, etc. and you have one of the following:
bad clutch switch
bad ECU
some sort of trouble between ECU, pulse module, the key switch, etc.

If pressing that relay does NOT make the starter and engine spin, you have:
bad starter
bad starter solenoid
super weak battery
really bad battery cable connections, etc.