Wing Commander only detects button 1 and 2 (or 0 and 1), usually the trigger and another button, the others can only be mapped either with Dosbox or with a mapping software, Logitech has one specifically for its products.
My advice is to go the logitech site and download the software, use it to map the other buttons on the stick, leave the dosbox joystick settings alone if there aren't issues.

From what I can understand the software also allows to assign values to unused axles, you could probably assign the keys for roll to the twist axis for example, and from what I see you should be able to map almost everything to the stick as WC1 doesn't really have that many keyboard commands compared to a game like X-wing.

Of course you can also do away with the default button 2 map (clik launches missiles, hold uses pitch to speed up and down and turns yaw into roll, useful for old two button, two axles joysticks but annoying on modern sticks with twist axis, throttle and buttons galore) if you can assign the roll to the twist rudder and just map it to the keyboard command to launch missiles and use other buttons to speed up and down like I did.