Clearly you, unlike many others, are aware that corned beef and cabbage is in no way a national Irish dish! That also annoys the hell out of me. Just like “soda bread” with baking powder in it! Argh!
Clearly, I am too easily annoyed. 
I’d go for a nice big dish of colcannon, myself, if I were still able to eat potatoes. Perhaps lack of potatoes accounts for my crabbiness, LOL.
OP, your behavior is exactly what is meant by being an ally. I don’t think you were wrong to address it. I assume (here I go again) that if this was your friend, you said it in a way that is worthy of your friendship. I hope your friend calms down and you two can have a healthy discussion about this.
^Irish (and Italians) certainly DID face a lot of discrimination. https://www.thoughtco.com/immigrants-overcame-discrimination-in-america-28345
^In fairness to modern Irish, the Irish International Immigrant Center does a great job helping all immigrants in Boston.
MODERATOR’S NOTE:
I like me some green beer as much as the next guy, but what does this have to do with the OP’s question? Let’s move on, please.
Update- I thought about it and maybe the way I told her was less diplomatic than it could have been–and it was by text so it’s hard to convey tone. Anyway, I told her I was sorry if I upset her (although I did not apologize for bringing it up because really, the picture needed to come down). She’s calmed down a bit although I do think I lost her as a Facebook friend. On a positive note, I think she’ll be more careful posting on social media. As for getting her to understand why it’s offensive- I don’t think that happened. She thinks we are ridiculous.
I can understand why she is confused. Some stereotypes seem acceptable to many people on this thread (see the Irish discussion) while others don’t ( see the Mexican discussion) even if the same humorous motivation underlies both and no malice is present in either. Best to dress as an animal at Halloween these days so nobody is offended in sensitive timess.
There is a difference between “political correctness,” which limits rational speech, and insensitivity.
Example of the former : using the term “differently abled” instead of “disabled” because you might offend a person with a disability. This is patronizing nonsense.
Example of the latter: using a racially offensive term and calling it “telling it like it is,” wearing blackface, Nazi Halloween costumes, etc. People who do this are not anti-PC. They’re just jerks.
While I wouldn’t let my kids dress that way, in both the cases cited the offending actors were children. I’m not sure we can label them as jerks. Perhaps their parents are insensitive, but I don’t know that the young children were.
Good on you for taking the initiative to make the first apology. I will bet that after it fades and she rethinks it, she will be more aware the next time. When people are on the defensive, it’s hard for them to see what they did wrong, when they settle down, it’s easier to see what is offensive. No doubt she just viewed the picture as an adorable shot of her son, many years ago.
It is sometimes hard to catch up with what is okay and not okay. I’m old enough that I have some sympathy for the point of view that Halloween is about being transgressive and that, it would be nice if everyone would just relax, but I’m also progressive enough to believe that if someone, and especially if lots of someones, are telling me some stereotype is hurtful, I should probably listen.
I think you did the right thing, and I’m glad you tried to build some bridges with her. It may take a while for her to realize you were right, or it may never happen, but societal change often happens slowly until suddenly we all wonder what the fuss was about.
What i always find interesting that something that will offend someone in the US does not in another country and looked upon with pride. The cartoon character, Speedy Gonzales is considered to be in bad taste in the US but he is very popular in Mexico. In the US, he was seen as a bad stereo type of Mexicans. In Mexico, he was seen as the mouse who always got the better of the gringo cato.