Make sure you start by changing the control method away from mouse & keyboard. I don't think it should make a difference, but it might. There's a combo-box in the upper-right corner. Some sticks have pretty good defaults already programmed, but you may have to start from a generic joystick binding.
Assuming I'm looking at the right stick, it looks like you've got a good number of buttons at your disposal. I would probably assign one of the hats on the stick to power management (right: weapons, left: systems, up: engines, down: reset) and the other to targeting. I like to use up/down to cycle hostiles and left/right to cycle subsystems. Using one for headlook is not a bad idea, either. I prefer to use the accumulate method and assign a button to reset view. On my X-55, I use the knobs on the throttle to control my vertical and lateral thrusters. I find it very useful to have them on an analog control rather than only being able to use them full-blast. It's particularly good for docking.
You'll probably want a button for boost, one or two for defenses like chaff and shield cells, and one to change fire groups. Primary and secondary fire, obviously. If you use your throttle in forward-thrust only mode, you may want a button to toggle it to reverse thrust. And activate FSD. It's hard to tell, but I might have suggested more buttons than you have. You can set chaff and shield cells to a fire group, so those would be the most likely candidates to cut from the layout.
Most other features you don't necessarily need to get to in a huge rush. If you have buttons left over, quick access to the UI screens is good. Landing gear, cargo scoop and the galaxy map are things that you'll access quite a bit but not necessarily in a rush, so they can probably stay on the keyboard.
