On my mill I have encountered a burr building up on the back of my band, it doesn't happen often perhaps from a defect in the hardening process the band went through at the factory, and have taken the burr off with a sharpening stone. Yes, with the mill running very similar to Jay's description. I didn't notice the fine print in Jay's post, so I am taking it upon myself to put it out there. Use extreme caution when doing this, need I say more. I have 27 stitches in my little finger alone from getting tangled in my debarker, luckily not the band. My finger still works, it looked like cottage cheese with some pink mixed in at the time.
I do agree the rounded back would help. You can also get narrower bands than 1.25" I have never tired but I may order in a special 3/4" band for such cuts. Maybe less thick, too. It only need to make a few cuts now and again. It should hold up.
I would suggest if anyone wants to try cutting curves on the band try on sample stock first. The idea of this comes from two different directions. One, running a dull band and seeing it dive two inches and then coming back out of the dive while I was not attending the levers, I was off bearing by myself while the band was self feeding. Two, being impatient when sawing, taking a cut and wanting to rise out of the cut befor I get to the end of the log due to a trashy end on the log that will not yield and decent lumber. Oh, a third example, cutting long stock I have been know to start out cutting 2" thick stock and by the mid point or so, drop the cut to 5/4 or 1" for the other half of a 36'cut.