Does it sound idiomatic to say "Regretfully, I won't have the privilege of being with you that day, but I do wish you all good luck."
Solution 1:
To me (a 27-year-old American), that phrase sounds very, very formal. I could imagine a professor saying that to a student, or an employee of a financial institution saying that to an important client. But it sounds too formal for most business communication, and it's definitely way too formal to say to a friend.
For most business communication, I think I'd say: "Unfortunately, I won't be able to join you that day, but I do wish you all good luck!"
And to a friend, I'd say something like: "Sorry, I can't make it, but good luck!"