How can I minimize the damage of my PhD advisor's bad recommendation when applying for industry jobs?
You are worried to much.

My PhD advisor also has several connections in HR. This can also be
damaging if a company background-checks through the university HR
department.

First, HR is not allowed to say anything bad about you with no clear evidence (for fear of being sued). They will just give minimal information, and that's it.
Second, background check is often done by a third-party, who is only interested in ticking all the boxes that you declared. They are not interested in hearing some random stories.

I do not know how references work in industry, but I can certainly
exclude my advisor in any list throughout a recruitment process.

For non-research positions, nobody care about references.