Thank you! I did mix up the inequality on $|S_n|$, as you noted.

@LiviuNicolaescu, Thanks for the reference, too! That made everything very clear. It's also a very nicely written book. If you want to put this into an answer, I would approve it.

@BertramArnold Thanks for the reference! I hadn't heard of supermanifolds before, but I'll take a look at them!

That's a fair question. The integral is just a real number. For the second part of your comment: maybe it all comes down to whether you can pull back a section of a differential pseudo-form. I don't know if it's possible either...

The integral of a section of $\Psi$ (as far as I know) is only defined over $M$, or over subset of $M$. In this case, the intergral is a real number. That is an interesting point about restricting $\Psi$ to the boundary. But, does the restriction of the pseudoscalar bundle to the boundary canonically give the pseudoscalar bundle of the boundary? If not, I don't see how you can integrate a $(n-1)$-form on $\partial M$ and get a real number answer.