So in theory, Dawn Blade should be something that significantly improves Hawthorne's damage potential, and could make a Sweeping Strikes Hawthorne potentially good.

I believe this interaction was intended to work. But the wording on the cards makes me think they do not:

So can units at range 1 of Hawthorne be considered engaged with him or not? The way I read this, Dawnblade should not be able to be used with Sweeping Strikes...

Any other interpretation of this out there?

If your front edge is treated as touching the enemies front edge, then you are engaged with them. That's my understanding.

34. of Rules Reference "Engagement"

I read it like that, that you are not engaged without the sweeping strikes but because of it you are engaged, during the attack only

The "not engaged with part" is because you can't treat your front edge as touching an enemy who is flanking you. If that is the purpose, then from a rules-as-intended point of view, they should work together because touching makes you engaged. So as long as they are not touching Hawthorne's base, you could ignore the part about not being engaged with that enemy. But in general, we like rules-as-written around here, so I hesitate to rule on this interpretation.

I do think @Athelin has a good argument, and I would go with that.