It's a shame you're not allowing walk-ups to play the event as the solution is simple:
Also, having run such an event before, I'll explain why each of these don't work.
* Let walk-ups use a disc from someone who pre-registered for the event.
-players were given months to pre-register. Signs posted at courses, email threads, facebook, pdga board. EVERYONE IN THE DISC GOLF COMMUNITY KNOWS ABOUT THE ACE RACE. Follow the rules like with every other tournament. Players showing up the day of the event should not get to participate without paying the entry fee. Not fair to those who paid.
* Order packages for all the walk-ups after the tournament has concluded.
You can only re-order a minimum of 10 packs. Last year we had to order 10 full packs for the 2 walks up that forced their way into the event. Then I was stuck trying to sell 8 ace race packs for months after the event. Trust me, the dicscs aren't very popular when the market is already flooded with the disc. AGAIN - FOLLOW THE RULES, sign up ahead of time and you don't need to walk up.
* When packages arrive, give the walk-ups their items less one disc.
-see above why it is not a good idea to re-order, and the problems that come with reordering packs. Also see above with following tournament rules.
* Give the one disc to the person who graciously let someone play the day of the event.
-see above with the minimum ordering of 10 player packs.
Not everyone reads disc golf message boards or visits the PFDS web site. I guarantee there will be walk-ups the day of the event who had no idea of the pre-registration cut-off. Hopefully, those people turned away will be league players or club members and not somebody relatively new to the sport who might be trying organized disc golf for the first time.