Pain in the as

No peeking, now -- what does the hed mean? More particularly, what part of the hed does "as part of dispute" go with?

The logic of news pulls you toward "fire," but I don't think the grammar gets you there. Here's the print lede:

Ferndale police have a 26-year-old Hamtramck man in custody after a fire early Friday in the 1600 block of University that investigators said was the result of a family dispute. Fire crews said the blaze was intentionally set.
Once it's clear that there's a clause after "fire," the smoke clears. There's room for "set" -- "... fire set as part of dispute" -- but that brings another problem into view: the lack of attribution. When the stuff you want to say doesn't leave room for the stuff you need to say, the best answer is usually to say less stuff, as in the online hed:

Man in custody following Ferndale blaze

... but unfortunate stuff happens to the lede:

Ferndale Police have a 26-year-old Hamtramck man in custody after a fire early today investigators say was the result of a family dispute.

I'd say the "that" in front of the relative clause would be optional if nothing came between it and "fire." But with the time element (or the time and the address, above) intervening, you need to keep the "that." If you haven't omitted your day's quote of Needless Words, hang on for the obligatory last sentence:
No one was injured in the blaze.