I want him to be more stable.
For the past couple months he says he's going to find a new therapist but haven't done so yet. The motivation isn't there for him I guess.
Motivation is complex for someone with BPD. If he is emotionally dysregulated, his problem-solving abilities will be nearly nil. If he is in a cycle of self-invalidation, the sheer weight of shame may make him feel hopeless.
It is baffling to those of us who can manage to pull ourselves together and get things done, even when stressed, anxious, flooded with emotion. I try to remember physiologically what it felt like during the most humiliating experience I've had, and multiply that. It helps me make sense of baffling BPD behavior.
What do you think about talking to him when he is emotionally regulated (it sounds like he is quite aware when regulated) and learn to set some boundaries? He probably knows he is not stable enough to take care of your son alone, and feels a lot of shame about this. The key is to address how he feels without making him feel more shame, and that's a skill we try to practice here.