In the PS2 version, you can control two ships with one controller, but when I use the config utility, it won't let me assign anything to any of the Joystick P2 controls.

This is strange, I just tried and succeeded in configuring my right stick to work for 2nd player, enabling me to play the game in Double Mode without issue.
I know you posted this message here a long time ago, but have you tried to reset & back to default the settings in the config utility ? (with the buttons "Back to Default" and "Reset all Settings"). I believe I had the same problem first, but used this two buttons to go from scratch, and after, I was able to properly configure buttons the way I wanted (I'm not sure, but it seems I had to press some buttons of the gamepad while selecting one key I wanted to configure to let the config utlility detect the gamepad).
I hope this will help :)