Here are what keys I mapped using the Logitech Profiler Software version 5.10.127:

R1 = Spacebar
R2 = Right Sidekick
L1 = Change Firing Mode
L2 = Left Sidekick
Up Directional Button = Up Arrow Key
Down Directional Button = Down Arrow Key
Left Directional Button = Left Arrow Key
Right Directional Button = Right Arrow Key
Up Analog Stick = Up Arrow Key
Down Analog Stick = Down Arrow Key
Left Analog Stick = Left Arrow Key
Right Analog Stick = Right Arrow Key

When I play the game R1 fires correctly but R2 moves the ship forward, L1 does nothing and L2 moves the ship backward. The keyboard commands in the game are set to default so I have no clue as to why this is happening.

I had tried this in Windows 7 but unfortunately did not work for me.

I did however find a way that worked using XInput mode and native DOSBox joystick emulator without the whole logitech profiler thing (which never seems to work for me -_-)

Edit ...\Tyrian 2000\dosboxT2K.conf

create the following section and save.

[joystick]
timed=false

For me, the joystick works flawlessly now!
Hope this helps.