The thing is - I can access the joystick and buttons if I REALLY need to, which I believe might occur at some point. I just simply wasn't ready to open it in the first few days after I assembled it. I'm not really looking for software fixes - instead I'm interested in Windows registry fix, which changes axes on a driver level. As a workaround I installed the xpadder as it seem to do the job.
Also, please keep in mind it is my very first bartop arcade, and I'm about to build the second one soon, so I will certainly learn on my own mistakes.