Parker,
I completely understand your desire to have that experience with your dad.
That being so, and Aug 16-18 being a Friday to Sunday, get thee to St Andrews Thursday afternoon/evening after you get done working in Hoylake.
The height of tourist season and on a weekend, the ballot will be competitive, so you want to give yourself the best possible chance to get on TOC with your dad. Of course you can likely get on as singles any day, but it would be a shame to go all that way and only play it in different groups.
Aside from that, enjoy St Andrews, play some of the small courses nearby like Crail (Balcomie), visit Anstruther for dinner from the fish bar... spend some time really immersing yourself in a special place.