Funny story: When my dad died, the guy I was dating ghosted me (basically, I got the text saying that I needed to fly home right now, and I told him (he didn't even offer to give me a ride to the airport), and after that pretty much never heard from him again after dating for several months). And you know what, I'm glad he did, because that guy was a piece of shit. Nor did it make my situation any more difficult. Your situation is very different, obviously, but what I'm saying is, if something needs to end, it just needs to end. It doesn't sound like he's really leaning on you for support anyway, and while I admit it sounds bad from an objective perspective, you're not leaving him because his dad died and you just don't want to deal with it but because his shitty behavior is becoming very apparent.

Also, it's been a month. Yes, it's still very soon, but it's been four weeks that have felt like four months. How much time is enough time? It will probably never feel right. If you need to do it, I'd say just do it...but be prepared to accept that he may eventually try to make you seem like the bad guy to his friends/family. If you don't share a social circle, that should make it easier...

Of course, mine is just one random experience, and it would have been very different if I'd thought things were going fantastically with that guy beforehand, so I think a lot of it has to do with your own perception of how things were and whether you think his behavior now is just his natural behavior coming out or if it's a result of the circumstances.