I happen to have the very same joystick, and have been using this configuration for a while:

Button 1 (trigger): Fire Primary
Button 2 (missles): Fire Secondary
Button 3: Launch Countermeasures
Button 4: Target Ship in Reticule
Button 5: Target Closest Attacking Ship
Button 6: Target Next Escort Ship
Button 7: Target Nearest Hostile Bomb/Bomber
Button 8: Afterburner
Button 9: Target Next Live Turret
Button 10: Target Next Subsystem
Button 11: Cycle Forward Primary Weapon Bank
Button 12: Cycle Secondary Weapon Bank
Joystick Hat Up: Target Next Closest Hostile Ship
Joystick Hat Down: Target Previous Closest Hostile Ship
Joystick Hat Left: Transfer Energy Laser --> Shield
Joystick Hat Right: Transfer Energy Shield --> Laser
Turn/Yaw Axis: Joystick/Mouse X Axis
Pitch Axis: Joystick/Mouse Y Axis
Bank Axis: Joystick rX Axis
Absolute Throttle Control: Joystick Z Axis

The joystick's fifth axis goes unused in this particular game. I would use it for side thrusting if FSOpen 3.6.12 allowed me to assign left/right thrust to an axis control rather then just hat buttons. I personally find using the hat to look around to be less then useful, especially considering the sheer amount of information the game's HUD gives you.

Equalise Shields, Duel-Fire missles, Target Nearest Friendly, and energy management are not assigned to a joystick button, but I still use them a lot.

Occasionally I re-assign controls when playing user-made campaigns as needed. I still have not found a satisfying way to add side thrust/glide to this control layout when playing campaigns that support it.

Edit: Buttons 11 and 12 would be occupied by 'equalise shields' and 'target repair ship' if the joystick layout wasn't so unusual. But the locations of said buttons makes me want to assign them to switching weapons instead.