Captain America gets a great ability on his injured side: but how many dice would you need to roll to mirror it?

Short answer: lots. It’s a really good defensive ability. There is only one mystical attack in the game, so you are nearly always rolling 4 dice with blanks counting. Running the numbers on it, that makes it better than 6 dice: the amount he can get by paying for his Vibranium Shield power on his healthy side:

Add in that he can still activate Vibranium Shield on his injured side, and it gets really silly.

Better than 9 dice on defense. That’s… impressive.
So what can you do? Well, because of the wording of I can do this all day, you only get the benefits against attacks. Throws aren’t attacks, so throwing terrain or characters at Cap is the best way to take him down. It will also get around his Vibranium Shield, which also requires an attack to trigger it.
Otherwise you may find yourself facing the equivalent of better than 9 dice in defense…