This problem occurs for EWJ1, EWJ2, and other DOS games with a similar gamepad setup, like Rayman Forever. The gamepad calibrates perfectly, but it's always missing two buttons for some unknown reason. I'm using a Logitech universal and it's surprising to see it not work. For example, I go to the EWJ2 gamepad calibration and you "Whip the Pigs" (which is amusing of course), and it calibrates successfully. BUT when I go to Joy Key Setup it will only let me set up the A and B keys and it leaves out the important C and D keys. Without those buttons I can't even shoot anything.
Does a solution exist? Or should I just go back to something like Xpadder?

I Don't know if you ever found a solution to this, however one thing I would check is the type of joystick that is set in dosbox's config. Check the manual for possible values.

I actually eventually decided to just use my PS3 controller with DS3 tool. For whatever reason the DS3 drivers work with almost everything I've tried it on.
Don't know why I didn't try it earlier.

I've used JoyToKey to play EWJ with gamepad.