Most car alternators should be able to supply the necessary current to the player. The only time that the alternator output should be an issue is when you've got a lot of other aftermarket stuff plugged into it, such as very-high-wattage amps, neon lights, etc.
With that said: On my two-drive player, my display goes blank on hard drive spinup and I get the battery icon when it's in the car. But only if my engine is off or (sometimes) if I'm just starting the car. If my engine is running (and the alternator is doing its job), then I don't have the problem.
I wonder...
Is it possible to change the bootup procedure in the kernel to "stage" the hard drive spinups? Spin up disk 1 then disk 2? Any of the kernel hackers here reading this?