I am so sorry to hear about this incredibly scary experience with your precious LO. Babies are so hard to figure out as it is and when severe allergies are on top of it all, it gets so much harder. I agree with you that it sounds like allergies is what your baby is experiencing and I also agree that waiting doesnt make sesnse when your Lo is suffering.
My DD also had allergies from the beginning that were very hard- no head to toe skin conditions then (we've had that more recently at 20 MO) but endless crying and arching back and crying post-nursing, tons of nightwaking, inability to sleep, mucusy and sometimes bloody poops etc.
If it were me I would go straight to elimination diet and get really simple with brown rice and other foods that are non-reactive. I'm no expert in elimination diets so I would look elsewhere (allergy forum would be a good place to post this too!) but I found with my daughter that dairy, soy, wheat/gluten, peanuts, eggs, chocolate, banana, blueberries, strawberry and corn were all issues so they are out of both of our diets now. I started with diary and played around with broccoli, cabbage etc. that everyone recommends but elminating dairy, wheat, soy and corn helped my daughter to stop crying constantly, get some sleep, start to smile and enjoy herself, etc.
I've been reading Your Child's Food Allergies by Dr. Scott Sicherer which I believe is supposed to be the go to book for figuring out allergies. He talks about how studies show that most breastmilk does contain the proteins of the foods the Moms eat so elminating the big allergens from your diet immediately could make a big difference.
Of course, I remember reading on kellymom.org that when you eliminate you need to figure 6 weeks from time of elmination to complete result because it takes up to 3 weeks to eliminate say dairy protein from your system and then another 3 for the baby to eliminate it. This said, we found differences within a few days often times.
I would also look at non-food allergens in the household and get things as hcemical-free as possible. Because your babies skin is so irritated, I would make sure ;laundry detergent is chemical and frangrance free and super mild, stay away from softeners, dryer sheets etc andlook at all skin, bath, shampoo etc. products that go on your babe, as well. We use california baby and it is pretty allergen free.
I would check with Moms on the allergy forum about the tests. I think these aren't too accurate, especially for little ones but they have more info over there. I would also consider checking into specialists as well as alternative therapies to consider all options and help you feel like you are getting to the bottom of it all. Mama wisdom is big so trust your intuition if you need to get specialists involved. I would hesitate on big time poking and prodding myself, but you and your DH can figure out what is necesary.
I wish you luck in figuring this out and strength and patience as you navigate your way through. We are still figuring it all out with my 21 month old and its complicated. The first step is getting this reaction calmed and hopefully elimination diet can get this going. Good luck!