• Is this kind of push–pull dynamic common during splits?
It's not only common, it's 100% going to happen and is one of the main characteristics of someone with BPD. They don't know what their emotions will be from one minute to the next hence the 'I want you /I don't want you' actions, which are devastating when they suddenly start appearing in what was, until then, a seemingly perfect relationship. All par for the course and everyone here has experienced it.
As others have said, it isn't going to get any better and will usually get worse over time. It's a script which inevitably plays out. The only thing to do is to decide whether you're able to try and handle this type of relationship - and it can be very draining and damaging, both mentally and physically,
In view of the short duration of your relationship, it might be better to just put it down to experience and move on. None of it was your fault; you couldn't control their actions nor could you cure them.
My own BPD relationship was 4 years duration and her first breakup with me came after a couple of months. I realised then something was wrong with her but I continued to chase and engage for all those years, only for it to inevitably end. Looking back afterwards I could see the relationship was toxic and was never going anywhere.
I truly wish I'd just let go after her first outburst then I'd have saved myself those years of greif and conflict.
Best wishes whatever you decide to do.