I'm in chapter 5 right now, and I'm worried that if I don't re-beat all of the old bosses I might miss some bit of story or experience. But they are getting a little boring, and most of the ones I've beaten so far have only given job experience. Should I beat all of them before moving on?

If you don't already know, you'll be fighting the bosses over, and over, and over again (5x or something asinine like that). Some of them provide more insight into character backgrounds, some are interesting alterations of the enemy's persona, and some are just copy+paste. It's not terribly important, and definitely not necessary, but if you're going for the true ending (don't break the crystals!) you'll need it. After the 4th time, all the bosses get really hard. It forces you to think of tactics beyond your comfort zone and try new things, and overall it's a rewarding challenge in and of itself when you get to take down four of them in a single match. Fighting them later on when they're ridiculously overpowered is a requirement for getting the true ending anyway, so think of all battles as practice. You can skip the 1st, 2nd, and even 3rd boss rematches of you want, and you'll still be prepared for their "final" forms