I was waiting for Eevee to come back from military to post in this MU so that we could run the BA/Joker MU before I posted. That then asked his opinion since he has played the MU as Joker against more notable BAs than I.
Basically here's what we came up with:
Pros:
-A lot easier for Joker to approach BA than for BA to approach Joker, making the health lead important.
-Oki game is way better for Joker.
-Not sure what you guys are talking about lightning cage messing up BGB setups. Depending on when you jump over BA, you can change his inputs at any point, making it very hard for him to lightning cage so long as you don't use the same timing every time. On top of this late teeth will catch his wakeup in it's recovery. BA's best bet to get out is a well timed backdash.
-There are a few ways to consistently stuff Lightning Cage, but many of them are very strict on timing or spacing dependent (For example D2>Crowbar will always beat it, but only at a certain range). On top of that most ways to stuff the wakeup make the knockdown situation easier to block.
-With gun held out, Joker can punish Divekick, lightning strike, black magic, dashes and bomb on reaction before they hit.
Cons:
-Can't really punish divekick on block unless it's done really high (or with flower if you have AI level execution). That said, jump backs and MB B3 will beat it out.
-BA actually has some very nasty setups in the corner with the bomb making his corner game dangerous as well.
-Lightning Cage is a very good wakeup.
-The timing is a bit strict, but Black Magic can punish a whiffed gunshot.
Miscellaneous notes:
-While BA's pressure does more damage thanks to trait, it is very straightforward and easy to block/pushblock. On the flip side, Joker's pressure is a little less damaging, but harder to block and can lead into more/back into itself on bad guesses. They both put you in + frames for days.
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AL Eevee what am I missing?