If you don't need to print the invoice with the correct information (under the correct customer), and only transfer that balance, that is easier (/M credit and /M invoice, using a suspense GL account).  I would not do this if you are invoicing products (especially lot / serial items) which may need to be later traced back to the correct customer.

You don't need to create an SO for re-invoicing, and you'd really only want to do that if SO history means something to you (or if there is something left on back order).

The most complete fix is to credit the "bad" invoice fully, then re-create the invoice correctly.  For item costs related to FIFO / LIFO valuation , the costing amounts (CM vs IN) may be off by a bit, but there is no control over that.

For audit purposes, the proper procedure is to credit the invoice and re-invoice on the correct customer account.  It will also be the easiest way to figure out what happened a year or two from now, when no one remembers.  Sometimes the best way is not the easiest way.