You know, I think you probably could read just Breaking Dawn and get the gist of what you need to know to follow it at least by the end of the first half.

First book, Edward+Bella=4eva, established in wedding sequence, check; Cullens' relationships to each other, established in wedding sequence, check; some vamps have extra powers, covered throughout, check. Second book: werewolves vs. vamps 4eva, covered in pregnancy sequence, check; edward is wildly overprotective, covered in pregnancy sequence, check; Volturi really creepy mean Vampire Lawgivers, covered in wedding and trial sequences, check. Third book: newborn vamps super-strong, covered when Bella wakes up, check; Volturi really creepy mean etc., see above; vampires are lifetime pair-bonders, check.

All you really wind up losing is the James/Victoria stuff and the high school stuff, and really, does anyone need any of that?