Not sure why I didn’t look into this sooner, but the app Documents by Readdle will allow you to download audio to your device via a built in web browser with download functionality. Was just testing it on my phone, it’s really going to be about a thousand times better on the iPad but the good news is it looks totally free and seems to work pretty well (as a test I grabbed a break beat from Phatso’s Drum palace or whatever, worked just fine). You’ll still need something to convert to appropriate format in some cases but hey, at least you don’t need a computer to do all this stuff.
Now the million dollar question - AudioShare USED to have a web browser built in for this same purpose, but Apple made the dev remove it. Why does Readdle get a pass?