Should I continue breastfeeding?

A doctor can not diagnose a milk allergy just by looking at your baby. If your doctor made this diagnosis and then did nothing further, that is not the standard of care. You should be referred to a pediatric allergist for a proper diagnosis.

Your baby arching his back and feeling discomfort is more consistent with reflux than with allergies. Although allergies can indeed cause nausea or vomiting, typically there would be other symptoms present as well. And if you have already removed all milk protein from your diet (that would include milk and also all dairy including foods processed with dairy proteins) then it is not an allergy to milk protein that is causing the trouble.

Weaning your baby rarely solves any problems, and often introduces new ones. Babies commonly have digestive distress of some sort with formula, because it is harder to digest than breast milk. If your baby has reflux, it would not be resolved with formula. If your baby had an allergy, you could avoid that food and fix the trouble, so no need for formula there (which might just introduce a different allergen). Go back to your doctor, or seek another opinion from another doc.

A "milk allergy" needs to be diagnosed by an allergist. It's not an allergy to breast milk (that is rare).

I would seek another doctor. The arching the back sounds more like reflux. You could also see a lactation consultant for more breast feeding advice, and make sure there are no breast feeding issues. Are you emptying the one breast completely before you switch?

If your pediatrician simply said "it's a milk allergy", with no further recommendations , I encourage you to find another doctor for your baby.

If baby is still reacting after you removed all dairy from your diet, then he doesn't have a milk allergy.
There are many things that could be going on, but none of them will be improved by weaning baby from the breast.

Formula is actually likely to make it worse. Go back to the doctor or get another and find out why he's doing this.

Continue breastfeeding.

I highly suggest you see an allergy specialist if an allergy or sensitivity may be present. Regular doctors are not qualified to be making those diagnoses as it is not what they specialize in.

If a milk allergy was present, you could avoid it by simply not eating milk products while you continue to breastfeed. He would not be allergic to your breastmilk, he would be allergic to the lactose/milk protein making its way into your breastmilk from the food you eat.

Get a second or third opinion before making any decisions.

I would talk to the doctor about what is causing this before making any decisions. It may be another intolerance and you may be able to keep nursing if you cut something else from your diet.

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It could be an allergy/sensitivity to something else. Speak with the doctor before doing anything. You may be able to continue to nurse if you just cut that thing from your diet. And sometimes, as babies age, they become able to adjust better to certain foods and sensitivities disappear. So you may be able to add dairy back into your diet when you take that other thing out.

I it becomes evident that there is a lot of stuff your son is sensitive to in your diet and a complete diet overhaul is needed, it may be more realistic to just switch to formula, which you should not feel bad about. Breastfeeding should be enjoyable for Mom and Baby and it sounds like Baby isn't quite enjoying it at the moment due to discomfort after feeding.