If the doomsday does it right they are alternating between using mb shoulder and them trait after knockdowns so he is always in your face with tons of armor. You can't zone because of trait and mb shoulder. Set ups don't work for the same reason. I can't get damage because of his trait. I can't find consistent way to keep him from activating trait whatsoever. If I pushblock I get a mb shoulder for my troubles or a trait up.
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I should preface this by saying I don't have a whole lot of match up experience versus Doomsday, but some of my thoughts so far:
I actually fight Doomsday the same way with MMH as I do with Hawkgirl, a lot of spacing/moving in and out alongside pressuring him before he can get his stuff started on you. I usually use MB back 3 to blow up pressure, never really use meter for pushblock because it's not particularly good versus Doomsday and of course against Doomsday's trait MB back 3 doesn't work, but it can be used to check his other options like venom and of course supernova plus jump ins for decent damage.
For trait, try to find strings, pokes etc that recover quickly because from what I understand of his trait it reduces hitstun, but there is still hitstun that he suffers from moves. Other possible options are to try to wait it out, but of course he gets in rather quickly so that's not necessarily the best thing you can do.
Martian Manhunter's style of zoning doesn't seem to be particularly effective against grapplers and armor heavy characters so he has to play more footsie oriented and I think trait to outrange Doomsday can be a huge help offensively and defensively to compensate for zoning being limited
MB pillar can blow up Supernova if you read your opponent's going to do a supernova and you're unsure if they're going to MB or not and you can do a teleport combo, which will at the very least discourage it.
Standing 3 blows up jump ins particularly with trait activated. I need to use it more versus Doomsday, but I feel like it would be really good for checking body splash jump ins, especially with J'onn has his trait activated.
J'onn probably does lose to him though. I'd like to play the match up a lot more with Martian Manhunter, I think there' a lot of counters that need to be looked into for him.