The objection that most have to the phrase "black lives matter" is exactly the same objection that most have to "all lives matter". That is, essentially no-one objects to the sentiment expressed in the words in and of themselves, but they are suspicious of the political motivations of those who use the slogan.
There is a relative minority of people that engage in what is called "vice-signaling". That is, they claim to object to a commonly held moral sentiment that they feel has been co-opted for a partisan political cause. I think it's probably a counter-productive strategy, but I think those people can be reasoned with if you can separate the moral sentiment from the political platform.