Disclamer: I am Asian, I am not from USA, I am part of LGBTQ.

They are not mutually exclusive, it is possible for people to support both BLM and the Law, all the while being part of LGBTQ and rocking a Dixie flag, it is very easy for one to fallen under the fallacy of false dilemma and stereotype.

The USA is a big country with a population of over 300 millions, regardless what the media may present to you, racial relationship between the Law and the Minority may vary depends on where you are from, and states often feels like different country, and because US is a federation, political policy are different between each states, in states where black people and police have good relationship, they may wear cloths with blue line or have blue line flag, I've seen it with my own eyes multiple times when i was visiting relatives in the USA, and I rarely experience racism in the south, while i have experience it the most in the West coast.

The media is always bias, often paint a incomplete picture, and loves to tar a large group with the same brush, one would say, US media is one of the biggest cause of bad racial relation, as media always exaggerate for the sake of clicks, and Hate and Controversial create the most Clicks.

Dixie flag, from my understanding, for many southerners, does not have the same meaning as the rest of US America, for them, it is a flag that represent US south, and many don't even know about/care of it's historically context, which is why many southerners is rather offended to be accused as racist or homophobic for owning the flag, as an Asexual, I personally know several people in the LGBTQ community who have Dixie flag, including a black guy and a Latino.