Jay, “Democrat party” has never made sense to me. If the point is that the Democrats aren’t democratic, then putatively they aren’t democrats, either, and “Democrat party” is equally objectionable.
If the reply now comes, “But I mean ‘Democrat’ only in the capital-‘D’ sense,” I will say, “Which corresponds precisely to the sense of ‘Democratic’!”
You don’t obliterate the root meaning of the word just by lopping off its adjectival ending, and a proper name isn’t an adjective to begin with. But hey, if it makes you feel better . . .