Absolutely, you can clarify your broth with egg whites. It is a very common practice.
However, as @JohnFeltz infers, you'll have to carefully plan ahead to work this into your cook plan. Clarifying broth with egg whites is a delicate operation, difficult to execute while entertaining guests.
If you separate the egg whites ahead of time, and set up your equipment ahead of time, you may be able to pull it off. Also, you'll need to consider how to keep your seafood warm for the 10 minutes or so it will take clarify the broth (a foil covered pan in a warm-ish oven might suffice).
Noting your recipe has
Shallots, parsley root, carrots and other suitable veg clarifying might not be worth the effort if you are going to reintroduce those back into the mix. They will surely reintroduce floating particles.
Here is a good video of someone clarifying a consommé with egg whites, starting with a cold stock. This video is someone clarifying a hot stock as you intend.
Note: unlike the videos, restaurants will use a stock pot with a spigot on the bottom to drain the broth through a cheese cloth.