Brewing a hazy beer like that I normally skip the Irish moss anyway so I think you're going to be fine.
Fining and the effect of flavor is beer style dependent... for the most part.
In nearly all styles that are generally clear to brilliantly clear there isn't a lot of loss of flavor components from clearing/fining.
In styles where some of the flavor is being contributed by stuff in suspension like yeast in a Hefe and white ale or in the case of suspended hop polyphenols for hazy IPA; in these cases you can expect SOME loss of flavors.
The same goes for mouth feel. Where some of these suspended things effect mouthfeel you'd expect a small drop when fining. But its minor.
I've said it before though that fining and flavor loss is not a light switch effect. The loss may be noticable if you had side by side beers to compare as in an experiment. But like in most brewing if you need to fine for clarity and you wish the beer had more of a certain flavor you'd more often have more luck figuring how to get more flavor (i.e. adding more of that ingredient) than eliminating the fining step.
Fining to me just doesn't drive that big of an impact on flavor changes. The recipe is a bigger control knob for that than is the process of fining.
(OK I'm off my soap box now.)