Can I add a second dishwasher to an existing power circuit?
David!  We have 2 DW in our kitchen and I ran a dedicated circuit for each to be safe. Also, that extra capacity might come in handy for an instant hot or garbage disposal.  Code is getting stricter and stricter on hard wired appliances...what you can share with plug attached appliances, etc. It gets ambiguous at times. IE: if my DW has a cord, is that a plug attached device?
Since you are doing a major kitchen remodel including running a new cable for the new range, I'd strongly suggest running another 20 amp circuit for the 2nd DW.
BTW.  Good job asking specific questions.