My nose also itches with a ff mask. I doubt in my case that there is any cause other than that I can't scratch it. Similar to why everything itches when you are in an MRI tube. You can't move so your mind plays tricks on you.
Consider not giving up on the nasal pillows. I too thought I was a terrible mouth breather. But now 3 years later I use a swift with no tape and no chin strap.
I never thought I would, but at the start I wrapped up my head to keep the jaw shut and taped the mouth shut. The air blowing up my nose made me able to breath through my nose like I hadn't for years. I was happy enough with that but over time I eliminated the tape and ace bandage and now am sleeping great and breathing through my nose.
Not that it will work that way for you but the point is don't count out the nasal pillow because you are a non-cpap mouth breather.