I had a similar reaction to a Swiss Miss hot chocolate brand. Caused severe joint pain and some other things. Must be the dairy, but I do have the experience that dairy in other forms doesn't cause the same reaction! You might check to see if there are other ingredients like corn or soy that might be causing a reaction.
Thanks for the suggestions about soy and corn but, thankfully, I don't have reactions to either of those substances. I do have to be careful about potatoes as I have reacted to those (stomach/intestinal pain), although it's much milder than my reaction to gluten. If I'm very careful to get potatoes that are FULLY ripened then I can usually avoid a reaction by not eating more than a small to moderate serving. If I get a batch of potatoes that aren't fully ripe...maybe still just a hint of green in the skins...oh, boy! From what I understand, there's more of the chemical that folks sensitive to nightshades react to in a slightly green, not-quite-ripe potato than one that's fully ripened.
Anyway, I'm now completely convinced that Breyer's either has a cross-contamination issue or they aren't being totally forth-coming in their ingredient list. I've tried 3 different brands of ice cream, 4 different whole-milk cheeses, heavy-whipping cream over sliced peaches, one brand of yogurt and a big dollop of sour cream on my gluten-free burrito. I didn't react whatsoever - not even the tiniest of reactions. Ate a small scoop from what was left of the Breyer's MCC ice cream and BAM - reaction!!! Therefore, I think it's a pretty safe bet to say that after eating all of that dairy and not having even one issue that dairy, itself, is NOT the issue. Despite what Breyer's FAQ's and ingredients lists say, I maintain and continue to maintain that gluten IS, indeed, showing up in their ice creams by one route (cross-contamination) or another (undisclosed ingredient). And, frankly, I don't really care which route it is because I will not be buying Breyer's ice creams again.