I have an old friend who has, over the past few years, gone through several cycles of depression, burnout, and isolation to the point of barely leaving his room. Often I don't find out how bad it's been until afterwards. I'll send him a couple of texts over a period of weeks or months with invitations to hang out and get no reply; eventually he'll get back to me explaining that he had another meltdown, we'll meet up and hang out like normal, and the cycle will repeat. He's especially unlikely to show up for any activity involving more than two people, even just coming over for dinner, despite the fact that my fiancée also really appreciates hanging out with him (he's described it as a periodical social phobia).
Although we've known one another for more than half our lives (we're both 30), our relationship is still developing in the sense that we share more and more of ourselves, unlike when we were younger and only talked about video games, comics and in-jokes (although I do still really value that shared reference base, and don't get it as much from my other friends). On the one occasion that I managed to coax him out on a walk recently (one of only two times in all of 2018), he told me lots of new things, e.g. about how events in his childhood have affected him long-term. Our relationship has also developed in the sense that instead of not answering at all, nowadays he'll answer (after a week or two) that he's currently in a bad state and won't be any fun to hang out with.
I'd like to see more of him, partly for my own sake and partly because I want to help, but I'm unsure how to proceed. He has other friends, but I don't know any of them, and I'm unsure of how many they are, how close (geographically or emotionally) they are, and how much they see of one another (by the sound of it, right now he's not seeing any other human being). Should I be insistent and try to communicate to him that I don't mind seeing him when he's in a bad place, or should I take a step back and give him space until he's ready to hang out?
Related question: when I do meet him, how much should I share about my own life? Sometimes I worry about talking about my career or personal projects, in case I make him feel bad about his own life seemingly not being as on-track by comparison.
(In case it's relevant: he has a long-standing association with the social services in re: financial and employment help, but I'm unsure how much contact he's had with mental health professionals.)