You are wise to be mindful about getting more involved with someone with BPD. Since you aren’t living together, don’t have children together and are not married, you haven’t yet experienced the closeness that all those steps would bring, which would exacerbate his BPD issues, since BPD is a disorder of intimacy.
Already you have seen wild mood swings, paralyzing fear of abandonment, and extreme depression. Suggesting therapy has been met with resistance and that is very characteristic of BPD, due to sufferers feeling intense shame and self loathing.
Even if he were to participate in therapy, many people with BPD (pwBPD) quit abruptly due to unwillingness to examine painful issues. DBT therapy has proven successful for some, but only if the person is extremely motivated to stay the course.
Something else to consider is the potential genetic component that could result in children with a personality disorder. You already fear that he is unable to take care of himself. How will he handle the added stress of financially supporting children and dealing with all the issues that arise being a parent?
It’s good that you’re being very thoughtful about how this relationship might evolve. Here is an article that might help:
https://bpdfamily.com/content/what-does-it-take-be-relationship