Actually, Ive used both. The Afterburner stick really isnt all that great. Its a little too clunky and slow to react. Was harder to move than other analog sticks.. which throws off your physical accuracy.
There is another typle of analog stick out there that is much better.
Instead of the springs being internal.. there is a metal spring that attaches to the bottom of the ball of the analog stick... and the other end attached to a metal frame that is about 6 inches deep. This keeps the force easy to overcome no matter which direction you go in... because the spring force in perfectly centered and pulling downwards.
Now... you may just be able to modify your happ stick to operate the same way.
Just remove the springs. Then drill a mounting hole on the bottom of the shaft. Stick a metal pin thru the shaft.. the attack the spring to the pin. Actually, Before that, drill a hole in the center of the pin. Then the spring will go thru the center of the pin's hole. While you could attach the spring directly to a hole in the shafts side.. it would eventually wear down and ruin the shaft. This way, if the pin wears down.. you can easily replace it. Also, it will center the joystick better.
Then you will just have to build a small 6" deep bracket that you will attach the bottom of the spring to. Make sure to use a strong thick spring about 3 to 4" and is an expansion spring.
Or, you could look for these joys on ebay. They show up once in a while. I think came from one of the later atari space fighter games.
The last option.. might be to try to build a centering spider simular to Sinistar's.
But mount it on top of the joystick rather than underneath. This method will give you the most amazingly precise results, if you can pull it off. I did it with a PC based joystick, and the added control was unbelievable. I used Bike tires as the rubber bands, and gromets to put the together. The happs analog's shaft is much thicker though... so comming up with the perfect way to make a spider for it will take some figuring out.