If you want the best advice, you need to give the whole recipe with all ingredients in weights. A list of the fats you used isn't all that helpful.
What reason did you have for using using less alkali than the recipe called for? I'm puzzled why you would make that choice, so could you explain, please?
Why do you want to rebatch the soap? If you need to ask whether to add the missing alkali when rebatching, that tells me that adding the missing alkali is not a motivator for rebatching. So I'm curious about your goals for rebatching.
To answer your question, yes, you can add the missing alkali if you think that's the right thing for you to do.
Here are two recent threads on rebatching that give people's opinions on this technique --
Can I rebatch this soap and
Should I rebatch or leave as is?
If you want to use activated charcoal to get black, then you need to add enough AC to get black rather than gray. I would normally use black oxide as well as AC to get a deep black shade. Since you're coloring the entire soap, expect your washcloth and the lather to be dark.