Haziness has nothing to do with yeast properties as such, certainly nothing to do with flocculation. The haze comes from the interaction of protein (mostly from the grist) and polyphenols (mostly from hops, especially southern hemisphere hops). Not the yeast.
And in fact there's an epic thread on HBT where someone DNA fingerprinted the yeast from Treehouse Julius, one of the real "hype" New England beers, and found what looks like S-04, T-58 and WB-06. Others have experimented with proportions and it appears that they were using 90% S-04 with a pinch of T-58 and WB-06, although it seems they may have changed yeasts since the time of the original fingerprinting.
So S-04 is not such a daft idea as it may seem - I think there's little debate that Lallemand Verdant is the dry yeast of choice for hazy stuff these days, but S-04 is OK. Maybe do the kit "as is" this time, and then see how it varies in future?