I've been working on the Doomsday matchup a bit lately. When he has armor on, mash d1 when within range, otherwise just pushblock - I think saving your meter to deal with trait in this way is absolutely key. I haven't found much else Batman can do, unless you want to waste bats just to keep him at bay for a small amount of time.
Without trait, he's a sitting duck for your crossup games unless he uses meter to get out. This is a match where my main focus with the bats is getting in, as you either get free mixups/pressure or make him waste his all-important meter to get out via MB shoulder charge or pushblocking.
Oh yeah, and don't get too greedy with jumpins after knockdowns. Wakeup supernova and AA grab will make you pay for that one. I believe the safest meaty attack is b1 and its related strings, as this game seems to have a bug where low attacks force the opponent to block BEFORE they can do a wakeup attack.
Finally, a VERY important recent change to this matchup which I just worked out in training mode: thanks to the patch nerf on splash hitbox, Batman can actually AA it on reaction now! He still can't consistently beat it, mind you, but he can consistently beat or trade with it if you have the proper timing. By far the best option for this is d2, as even a trade allows you to score a full combo (123 into b23 or b2 -> EX grapple etc.). Countering his splash with d2 will require some grinding in training mode to get the timing down, but once you get the feel for it you will be able to do this in real matches!
As for the Aquaman matchup, I'm still trying to figure it out as well. Never, ever try to fight him at midrange, and don't go for crossups in the middle of your pressure until you have him completely scared of your other options (so he'll be less likely to react with uppercut). It can be tempting to try and jump over his big pokes to land a bnb, but that's a guessing game not really in your favor since you run the risk of eating uppercut into his bnb. There may be some potential in whiff punishing with b3, but I'm not sure if you can hang outside his best range and still have it connect. The way I see this match atm is, hang back and keep trying to get in with bats. A good Aquaman will likely pushblock quite a bit, but just stay persistent until he can't or doesn't do that. When trying to land a combo, I tend to stick with b113 since it's safe and harder for Aquaman to escape with trait than 123 (since the first 2 hits are harder to react to). Unfortunately that's about all I have for now!
