Hmmmmm..... 7 days AND time for golf. That's tough.
I can't piece together a worthwhile ride thru here for you in 5 days of riding though maybe someone else can.
Your first instinct of finding a tour group may be your best bet. I have crossed thru this area twice as a subset of a larger trip and each time I was passed repeatedly by large passenger vans loaded with people and giant bike racks on top stuffed w bikes. I never got the story from a participant but I think they skipped riding anywhere that didn't warrent a picture in the brochure.
If you got golf clubs your lugging around I assume you're renting a car at least for some of the time. If I was advising a pal who was coming over from Oz on your schedule, I would tell him to (Heresy Alert!) forget the bike. If you are physically up to it I would tell him to hike to the bottom of the GC, then the next day drive over to Zion and over the next 2 days hike up Angel's Landing & thru The Narrows, then on day 5 go over to Bryce and take a hike there and then drive over to St. George and do your golf thing. Thats assuming you can still swing a golf club after the GC and 2 hikes in Zion.
Hopefully someone else has a better idea.