I'm sorry OP, that sounds like a huge mess and headache. HR is not handling this properly and the doc should have been told to leave you alone. There shouldn't be any question if they told you an employee would be fired but you're right, her position is probably affecting that outcome. Not fair but it happens a lot.
Stay calm, document, and touch base with HR to make sure they tell her to leave you alone. Let your insurance handle it with hers and push them for a comparable rental. Keep doing your job and being a model employee. Don't give them any reason to question your job.
Not that I'm in love with HR folks but the OP doesn't KNOW what HR has said to the Dr and how they are investigating the incident. HR can not and should not discuss any type of possible disciplinary actions of another employee.
OP, call the rental people and tell them you require a larger vehicle, comparable to what was damaged. They will clear it with the insurance company.
If approached by the Dr, walk away without comment.
Most importantly--CALM DOWN--no one was hurt and sheet metal can be replaced. Is the Dr a jerk? Heck yeah--