First, let me welcome you here. The peer support is solid, we have a tools and skills board to help you educate yourself, and we have many time-tested perspectives and strategies to help you in a variety of ways and circumstances. Others will chime in here shortly but I'll comment on a few issues you're facing.
That your son is saying he hates you is not due to you being away nor is it being in a different location. He is most likely echoing what he's hearing from his other parent. (Ponder that, have you heard her bad mouth you in front of him?) That is not surprising, many if not most of us have experienced being blacklisted too. BPD is a disorder where everything is perceived as all good or all bad, there is seldom gray zones in a pwBPD's perceptions and world views. So understand you're not really a bad parent nor a bad dad. You're being cast that way but that's not true.
Another perspective for you... A child having two homes is not bad. Why? His current home is one where her wide-ranging moods and demands define the home environment. That's not healthy for your child. However, in most cases it is not extreme enough for a child to be removed from a dysfunctional home. There's little or nothing you can do to fix her home. What you can do - over time - is establish your own home that is relatively stable, functional versus dysfunctional, healthy versus unhealthy. Your child will thrive there. The benefit is that in the years to come he will grow to recognize and appreciate the stability and trust of your home versus the ever-changing rules, demands and dysfunction of the other home.
This is just a general overview. For most of us, it wasn't easy. It took time, effort, grit and determination. It meant behaving as virtually angelic (to avoid us facing legal consequences for losing our tempers) no matter how bad or taunting the ex behaved. It meant smarter and better Boundaries (how we respond to the ex's misbehaviors). It meant informing ourselves of PD patterns and how to successfully deal with them and effectively counter them. It meant finding an experienced and proactive family law attorney to help us stand up as a parent. It meant seeking local support (
trusted friends, family and counselors) in addition to the collective experience and time-tested strategies found with peer support. It meant accepting that the adult relationship was doomed and ended but
being a parent is forever.

Edit: I have to add a final recommendation. Right now is a very unsettled time for you. You have no assurance how the future will turn out, whether good or bad. However, odds are the adult relationship is irreparably damaged. Therefore, until you can have long term confidence there is success with it, do not dig yourself deeper into expanding your relationship. (1) That means no more children. And you can't trust her if she claims she is handling birth control responsibly. (2) Don't buy a home, apartment, vehicle or any other asset or new debt with her. No added obligations with her or to her. Why? It would make unwinding the relationship that much more difficult. At the least, consult with your lawyer and with us in peer support before making any major decision that would make unwinding the relationship that much harder to accomplish.