I would rather not make it explicit since it would provide enough information for any colleagues of mine reading HN to connect the dots and be 100% sure who I am.
I was pressured to do something during that meeting, I refused with the argument that it wouldn't benefit any of us and I explained why. The person kept asking. I kept refusing. It was uncomfortable.
I know I've handled that particular situation in a bad way. I could just have done it even if it meant trouble in the future but I took a stance out of professionalism. It was a mistake, no doubt.
>I refused with the argument that it wouldn't benefit any of us and I explained why.
That's a bit of a lesson learned in terms of "Things not to do in your first week".
Aside from criminal activity, I can't think of any situation where you were in a position to understand things better than the person asking you to do the task, and even if you where, where it would be a good idea to make such an immediate disagreement vocal and public.
You should probably test the mileage of making an apology for trying to assert yourself over a senior person (regardless of whether or not you are in fact correct) in your first week before you do anything, really. I have a funny feeling that it might go further than you think here.
> 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.
Short of unethical business practices, a new contract employee really doesn't have organizational standing to say "no" to anything.
If you're a consultant and they're paying you for your breadth of experience, you can certainly advise a different path. And you can make a stay/go decision based on the organization's choice.
But as a contract employee, or any new employee, you have much less invested in the organization than the people you work with do. You are reasonably expected to defer to their judgement in organizational matters.
Over time that might change, but it doesn't always. In some companies, contractors are always second class citizens.
It sounds to me like your PO has specific expectations, and you flipped the "loyal and cooperative" bit in the wrong direction. She might be overreacting, or perceiving it as a challenge due to gender differences. It doesn't matter, even if you are ultimately proven correct, you need to be someone she can work with and trust, and right now you are not.
An apology might work. In my experience, nothing else will.
I was pressured to do something during that meeting, I refused with the argument that it wouldn't benefit any of us and I explained why. The person kept asking. I kept refusing. It was uncomfortable.
I know I've handled that particular situation in a bad way. I could just have done it even if it meant trouble in the future but I took a stance out of professionalism. It was a mistake, no doubt.