The Brotherhood clearly harbors anti-Stark, anti-Lannister sentiment, all based on their role in the Riverlands sacking. What was their opinion towards the stags? Had Renly or Stannis won the war and brought peace the Riverlands, what would have happened to the BwoB?

They’re not as hostile to the Starks specifically – “we mean your brother Robb no ill, milady” – but that changes after the Karstarks go rampant.

As to the stags, I don’t believe the BwB ever took a position on Renly or Stannis because they never came to the Riverlands.