If you want a book which is priced under \$30, write it yourself and put it on the internet. Then it's free! (This is not a quip or a dismissive comment: please do actually do this. I have done this sort of thing myself.) Among books that the evil empires of publishing put out, I used one for such a course twice and — apart from the price — it was pretty good:
Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand, Polimeni, and Zhang
I'm not sure exactly why you are against historical comments (nor do I know exactly what "mathematical perspectives" means in this pejorative context), but so far as I recall this book is fairly businesslike. (Added: I just processed the part of your question where you mention supplementing the book with material from Courant and Robbins. That latter book is all about perspective, so I guess the idea is that you want to avoid duplication of content, which is very reasonable. Sorry if I sounded overly critical before.)
I was most pleased with the treatment of logic and sets in the first two chapters: just about the right amount, with just about the right level of formality and sophistication ... to my taste, of course.
