What proteins have you tried??
Yes I recommend staying away from poultry(Turkey, chicken and duck) because they are all high in a pro inflammatory fat that causes inflammation and very low in the omega 3, but some Frenchies do well on duck but for your frenchie with the inflammation I recommend sticking to beef, goat, rabbit, lamb, Buffalo, kangaroo would be great and possibly pork. Definitely try the herring oil, bone broth and probiotics as that all will help improve gut health. If the vomiting is after a meal, I would give digestive enzymes(you can get probiotics with added digestive enzymes for dogs) and make sure to try bone broth too cause it soothes their gut and improves the immune system too and anti inflammatory.
But I highly recommend staying away from dry food(kibble) and canned especially for inflammatory conditions and picky eaters(Rusty won’t touch any kibble or canned lol).
Definitely stay with raw or a gently cooked.
And the Eye Stream will definitely help
Bovine colostrum will help for the paw licking and the bone broth too can help.
Also a suggestion:
Berry blends supplements(I use Four Leaf Rover as there are no preservatives and everything is organic and vet formulated), their Red Rover berry blend I use for Rusty to help with the eyes watering and paw licking. Berrys are a good natural source of quercetin which helps with allergies and the antioxidants help their eyes.
Since using their bovine colostrum and the Red Rover berry blend, no more paw licking and the bone broth is great for when they vomit and their overall health.
The herring oil will also help the inflammation and the itchy paws.
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