Looks like quite a bit of driving!
Good call by Ryan for Formby when near Birkdale. Formby has on site accommodation too plus another 18th hole course too.
Further up north East coast probably has more opportunities than the West coast. Suggest you don't miss the super-classy Muirfield and the unforgettable Cruden Bay.
Also don't overlook some of the lessor courses sited near to the big names you've already chosen, eg West Lancs, Hillside, S&A near Birkdale. Brora, Golspie, Tain, Fortrose near Dornoch, and between Cruden Bay and Muirfield there's not only TOC and Carnoustie, which you've already got targeted, but also Royal Aberdeen plus the likes of Montrose, Panmuir, Lundin, Levin etc. When at St Andrews don't just play TOC, play some of the others too. Also North Berwick, which isn't far from Muirfield.
Wherever you go, have a great trip, tell us how you got on and post some photos.
All the best.