I usually tag out when the shield starts and then tag again after the 2nd/3rd purple pulse, by then my main is only 3-4 seconds from being tagged in. Also, physical attacks dont work against Black dwarf. You need energy attackers like Cable, Iron Man, Scarlet witch
If you run to the very bottom of the screen the blast after his health bar doesn't reach you and you remain untouched. I've used this to keep my characters alive whenever I can't break his shield
Black Dwarf has insane physical defense. Use energy attack dealers.
That green bar can be broken if you deal enough energy damage (his shield is immune to physical damage) but even if you can't break it, you can use immunity or i-frame at the last second so you dodge the attack.