I don’t think it’s anything to do with case, just that the gender of the word is different between the dialects. ‘en’ or ‘a’ is just a suffix meaning ‘the’.
I'm not sure about that particular dialect, if it's case or not, but it's a fact that some dialects do keep the dative (though it's been disappearing somewhat recently). In addition to certain country-wide expressions which have kept old case forms, like "gå mann av huse", and "i live" (being alive), though I've recently seen so-called "journalists" in newspapers being unable to understand it and writing "i livet" instead, which has a totally different meaning.
It is certainly possible that a word may differ in gender between dialects. But the way dative is normally expressed in Norwegian dialects is that masculine words get the normal feminine ending and feminine words get the masculine one.
Oh right, I didn’t know that! I still don’t imagine that’s what is happening in this case; there are just plenty more feminine nouns in Nynorsk and similar dialects.