Can we separate justice from theology in Christianity? And if we do, is this right?

King's Charlottesville speech is about racism, Vietnam and oppression of the poor. If you add a few of the preceding lines of the speech to your quote this becomes clear:

May I say in conclusion that there is a need now, more than ever before, for men and women in our nation to be creatively maladjusted. Mr. Davis said, and I say to you that I choose to be among the maladjusted, as my good friend Bill Coughlin said there are those who have criticized me and many of you for taking a stand against the War in Vietnam and for seeking to say to the nation that the issues of Civil Rights cannot be separated from the issues of peace. I want to say to you tonight that I intend to keep these issues mixed because they are mixed. Somewhere we must see that justice is indivisible, injustice anywhere is a threat to justice everywhere and I have fought too long and too hard against segregated public accommodations to end up at this point in my life, segregating my moral concerns.

When he talks about these issues being mixed, he means things like:

• Poor men, and especially African Americans, were drafted at significantly higher percentages than rich white men. See Blacks and the Draft: A History of Institutional Racism
• segregation of facilities, services, and opportunities such as housing, medical care, education, employment, and transportation in the United States along racial lines
• The civil rights movement in the United States, a decades-long struggle by African Americans and their like-minded allies to end institutionalized racial discrimination, disenfranchis*m*nt and racial segregation

He is not talking about separating religion from politics. This is not how "separation of church and state" is interpreted in the US.

In the US, involvement of religious leaders and organizations is governed by rules for political activity that apply to all nonprofit organizations (including churches and other religious groups) that are exempt from taxation under section 501(c)(3) of the Internal Revenue Code. The prohibition on political campaign intervention did not become part of the Internal Revenue Code until 1954, when an amendment to section 501(c)(3) was introduced by then Senator Lyndon B. Johnson during a Senate floor debate on the 1954 Internal Revenue Code.

The First Amendment provides that “Congress shall make no law respecting an establishment of religion or prohibiting the free exercise thereof ...” Although the Internal Revenue Code prohibition against political campaign intervention may burden the exercise of religion to the extent that a religious organization must choose between the receipt of the benefits of tax exemption and intervention in a political campaign, not every burden on religious exercise is constitutionally prohibited.

Religious organizations are prohibited from participating or intervening, directly or indirectly, in a political campaign on behalf of or in opposition to any candidate for elective public office.

They are however, not prohibited from discussions of issues that are not linked to support for or opposition to candidates. The fact that candidates may align themselves on one side or another of an issue does not restrict the ability of religious organizations to engage in discussions of that issue.

Since you mentioned an observation on the Anglican church, in the UK the situation is constitutionally very different. The Queen is head of the State, the Armed Forces and the Anglican Church, and all UK citizens (whether they are Anglican or not) pay taxes for the Anglican church. The Church of England has a large endowment of £8.9 billion which generates approximately £1 billion a year in income (2019)1, this is their largest source of revenue. This is a constitutionally established state religion but other faiths are tolerated (although they do not receive the same preferential treatment). Besides the British monarch being the Supreme Governor of the Church of England, the church has actual political power: 26 bishops (Lords Spiritual) sit in the upper house of government, the House of Lords.