Personally, I'm very disappointed in the BHS for supporting the absurd response on social media. Given that Oliver Townend's horses were inspected prior to the showjumping, deemed fit to continue, and went on to jump well, I struggle to see how his behaviour can be considered abusive or of significant detriment to their welfare (at least, no more so than any other high level equestrian support).
That's not to say that I think he got it right - I can completely understand the verbal warning, and it strikes me as being a proportionate and sensible response. It wasn't particularly pleasant viewing, but I don't think his horses were under noticably more duress than many others which finished tired - and, indeed, I think they were a lot better prepared for their exertions than some others there.
I feel there's a very strong personal element to this, which devalues the argument against him - he should be held accountable for his actions, not for people's opinions of him as a person. I think that trial by social media is unhelpful and detrimental to the sport in general, and I'm disappointed to see that a couple of his sponsors have responded to that pressure (though others are standing by him, some rather vociferously!).