> I know I've handled that particular situation in a bad way.
This is a key realization.
Assuming you really want to stay at the company, you should speak directly to the PO if you can keep from being defensive/angry/frustrated.
To be specific, first talk to your direct supervisor/boss and explain that you would like to improve your relationship with the PO. You have to mean this, of course, but tell your supervisor that your only goal in talking to the PO is to make the relationship better and so ask your sup if it's ok that you chat with her and ask for any guidance on how to do that effectively (this is less about getting advice from the sup and more of a check in since s/he is responsible for you).
Assuming your supervisor buys into you chatting with her, then set up a time to chat with the PO. Specifically address where you think things went wrong, why they went wrong and propose how you think it can be better (use the argument you had, your role in it and how you should've handled it differently as an example of how you'll interact differently in the future). And then ask her how you can make it better. It might not be fair that the burden is all on you in this situation, but based on what you're saying about the PO's standing in the company it sounds like the burden is on you.
You absolutely can salvage this relationship assuming the PO is a reasonable person, but the key is the burden is on you to fix it. Talking to the PO won't magically make her a different person, but great working relationships can be founded on once-toxic situations. If the risk in talking to her is too great, listen to the many people who are saying stay quiet, do great work and find another gig.