It sounds like these people are exploiting you. They offer you very low pay, they strongly present a job that sounds like a certain amount of work, and then at the past minute switch it up for a role that involves significantly more work, and refuse to adjust pay? Someone (or more than one someone) are looking to squeeze as much out of you as they can get. You feel this is unfair because it is unfair. They are treating you poorly because they think they can get away with it... and, based on what your'e saying about how much you want this position even with the low pay and the increased workload, it sounds like they can.
So the first question is to re-evaluate. Are you sure you want this position? Why do you want this position? Do you really want to work for people who will treat you like that? If they are treating you like this now, when you have not yet started and it is easiest to leave, they are unlikely to treat you better once you're more locked in. If you do truly want this position, even so, then you should perhaps consider how long you expect or intend to keep this position.
That having been said... talking to the head of HR isn't likely to do anything one way or the other. If they really wanted you personally, they wouldn't be treating you like this. As such, he has very little reason to help you. If he signs off on your salary increasing, that's something he's going to have to answer for and explain. If he doesn't, then you either leave or stay, and it doesn't reflect on him personally regardless. On the flip side, if they've got you hooked, then the fact that you're complaining to HR might annoy HR a bit, but it's not like they're going to ditch you after they've caught you. Might as well try? Talking with the Principal isn't all that likely to help either, as he's not terribly likely to have control of such things.
Before you do any of this, though, you need to find out a bit more about how much teachers are actually getting paid. Are these people seriously lowballing you (most likely, based on what you've said) or is it that your idea of what teachers get paid is out of whack? If it's the former, it might work (though it's still quite unlikely). If it's the latter, you're going to get a lot less traction, and possibly a lot more annoyance out of trying anything.