Just out of curiosity, how many days per week do you squat?

What kind of mobility are you looking to improve specifically? It sounds like shoulders specifically? For bar placement for squatting I assume, given the context?

Biggest things I've found to help with that are shoulder dislocations (I use a band but you can use a tube of some kind). I do at least three sets using different grip widths, every workout. Having said that, if you're squatting low bar I think wrist and/or elbow stress is inevitable. Depending on how you place your grip you will be hurting one or the other over time. I position myself in such a way that it puts more stress on my elbows (since bending my wrists back really hurts my carpal tunnel). Elbow cuffs/sleeves help but only so much. Wider grip may become a necessity for me at some point in the future, but I lose upper back tightness so I've avoided this so far and just work on rehabbing my elbows instead. Some things to think about in that area.

That's all my experience, anyway.

Bar burn, where specifically? I've found that wearing a tight compression shirt and then wearing a normal cotton t-shirt over that helps a lot. But most of the bar that is contacting my back is the smooth portion so it really doesn't hurt me too much in the first place (I only do maybe 20 sets of squats a week though). If your bar is shifting around too much resulting in irritation, you could try grip/friction tape on the smooth part to keep it from moving around. Or alternatively, if the knurled portion of the bar is irritating you somewhere, you could tape that part up instead using a smoother, rubberized tape like you would find on a tennis racket (I think it's called overgrip tape or something?). Combination of everything may help.

Or just go with the pusssy pad.

Sure, your "buddy."