I hadn't been familiar with this usage until this thread came up the last time, so I was glad to be able to find it again, since there's not really an entry that corresponds to it in the dictionary.
I was reminded of it today when I came across a headline about the Irish elections. It was on the DW website and said 'Sinn Fein stößt bürgerliche Parteien vom Thron,' which made it seem like a catchall for all parties except Sinn Fein.https://www.dw.com/de/sinn-fein-stößt-fine-...
There seems to be a range of meaning, though. When you click through to the article, its headline names the two other parties instead, and in the text of the article, it says 'Die große bürgerliche Oppositionspartei Fianna Fail erreichte 22,2 Prozent, die liberal-konservativen Partei von Regierungschef Leo Varadkar, Fine Gael, erhielt 20,9 Prozent der Stimmen.' (Which looks like an ugly comma splice, by the way, but apparently that's not as bad an error in German.)
So does mean that in that case, the 'bürgerliche Partei' would be the more conservative of the two? Or vice versa?
Anyway, with that added context, and now the more I've been rereading in this and other threads in the archive, 'non-socialist' seems like a pretty good solution.
In a context already implicitly critical of socialism, 'mainstream' would also fit. But for readers who aren't aware of that, maybe explicitly mentioning the contrast would be better, so that the answer to 'mainstream as opposed to what' would be clear.