I think there are two problems here. One is strategic: the BLM organisation has a wide alt-left agenda, and the raising of this in the public consciousness in this way risks the possibility of diluting the mainstream support or their main constituency. (BTW, pretty surprising that BBC is running a 'Black Lives Matter' banner in its football, unless they are too stupid to realise that BLM is not simply a slogan but a political organisation).
Second is that that they are looking to conflate Starnier's (and the Governments') position on anti-Zionism (and which rightly condemns this move by Netanyahu) with Starnier's attempts to rid the party of its antisemitic, pro-Palestinian elements. Time to bring politics back to the Centre.