I'm snappy because I'm sick of the smug "I love my bed/ I need my sleep" comments from other people, all over the place, assuming they need/ deserve more sleep than those who have non sleepers, and because I find it crazy that somebody who could be having 8 hours sleep a night and a lie in once a week, alternating with her DH/P, is describing herself as exhausted - people who are exhausted sleep through kid noise if they know their partner is looking after the kids this morning, and people who are exhausted can get back to sleep standing up every chance they get, and people who are exhausted go to bed early to catch up every once in a while, to become less exhausted, if they know that going to bed will get them 8 hours sleep!
However I do have sympathy for early risers - I have one (not the same one who doesn't sleep at night). It is common sense to catch some early nights yourself - not every night, as the need for some evening adult time is important too, but a couple of times a week. It is also common sense to alternate lie ins.
Reward charts for being quiet may work, but you have to have reasonable expectations - if you know they will wake at 5am then you can't expect them to be quiet til 7am, but a simple, clear, mutually desired reward (like an outing they both want to go on) could be enough to incentivise them not to fight and be loud til 6am. Mark the desired time on a regular clock for the younger one if he can't tell the time - no need for a gro/ bunny clock by 4 really ( and my kids changed the time on theirs to wake the bunny)
Cheap MP3 players with headphones and their music or story tapes on them help pass the early hour before approved get up/ noise making time.
We used those strategies for our now 5 year old (who has always been a 5am get up boy) from age 3 and they worked pretty much immediately.
I have been to 2 different paediatricians for help with DC3 MrsB74 and been offered (and turned down) medical sedation - its not affecting his development, he's well ahead of his age and not showing signs of over tiredness, it's just how it is and I try to make sure it only affects me, though it is very, very, very hard, he will grow out of it.
From my perspective the OP is fed up of getting up early, not "exhausted and desperate" because of the choice she is making not to get more sleep when that option is very obviously available to her, and saying she is exhausted is a bit like saying you are "maimed" if you've pricked your finger with a needle.
This is not competitive, it is simply factual. Pointing out somebody is exaggerating and furthermore is actively choosing not to take up the obvious solutions to her own problem is not the same as competing.