It took ages for my ds1 to be diagnosed. GPs and HVs didn't seem to know a lot about it.
I'm guessing you have been doing some reading... My ds poos weren't especially green either, but they were mucousy. He screamed a lot (and I mean a lot!) and I was told he had colic. He had breathing difficulties sometimes, and went quite blue around the mouth on a couple of occasions. He kept being diagnosed with chest infections because of his breathing difficulties, and was on antibiotics 3 times before he was 3 months old- which almost certainly didn't help his little immune system!
My son's allergies were/are very severe, and let's hope your little one does not have that level of allergy to anything! Even though allergies are on the rise, most children are not like my ds, and I'm worried that Imight be saying things that will alarm you unnecessarily. Many grow out of their allergies in early childhood (this is especially true of cows milk and egg).
Perhaps the best way to find out is to keep a food diary, together with a record of poo and any other symptoms. You're right in thinking that the proteins in what you eat can sometimes pass into your breast milk.
Cows milk is an obvious culprit, so it might be worth cutting that out for couple of weeks to see if there's any improvement. Write down everything you eat and drink, because there are other common culprits e.g. eggs, nuts, soya, fish. (But don't cut them all out at once, like I did, because even though your milk will be nutritious, you will be knackered and a bit malnourished yourself. You should only be doing food exclusion diets under the care of a dietitian.)
If you cut out say milk, or egg, and see an obvious improvement after a couple of weeks, then you have the evidence you need to get a referral from your gp to an allergy specialist.
On the other hand, maybe your baby is mucous y because he has a cold and is swallowing snot and the reflux is "just" reflux... I know how hard it can be when your baby is poorly and you want some answers!
Let us know how you get on, and good luck to both of you