You can use a password to get by Chill Penguins stage without the dash boots. But the weapon upgrade is given to you at Sigma Stage 1, if you don't already have it from Flame Mammoth. So the only way to not get it is to hack.
Doing a minimalist challenge is impossible when you're fighting Sigma's last form. But you're limiting yourself to no wall slide, modifying your buster, and playing at 120 FPS?
BTW, good luck trying to get up the walls after you beat vile w/o the dash boots. Especially if you're going to not use wall slide. Not even sure if you can wall jump without sliding.