10 Year Anniversary Trip to Scotland with Stop off in London

You do not need a car in Edinburgh. And, like any big city, it is a liability that you have to find somewhere to park.
I suggest you do Inverness and St Andrews BEFORE Edinburgh. Dump the car when you get to Edinburgh. That way you are not doubling back north or repeating Inverness, so you gain a day!
Then take the train to London (4½ hours, and a lot more comfortable than flying). You then have:

Day 4: ? (Staying in Oban)
Day 5: Drive to St. Andrews (whole day drive through the Highlands)
Day 6: Golf at St. Andrews
Day 7: Drive to Edinburgh
Day 8: See Edinburgh
Day 9: Second Day in Edinburgh
Day 10+11: Train to London (Spend 2 nights)
Day 12: Fly back to Texas

You've just gained 2 days! If you don't want to spend them in Edinburgh + London, you can use one of them to stop off in York (half way between Edinburgh and London). Or you can spend more time getting from Oban to St Andrews.

I will highly recommend this Bed and Breakfast for your stay in Oban.

https://www.greystonesoban.co.uk

The home and the rooms are lovely, the bathrooms huge and very modern, the views from the window stunning, and the breakfast very very good. They have on-site parking for your car.

Agree with the train to London. Old Town Edinburgh is a short walk to the station 4 1/2 hours and you are in Kings Cross. If you bu in advance was like 25 GBP

As you're planning to play golf at St. Andrews, you're aware of the handicap requirement and lottery system for tee times?

For lodging I can recommend Old Station B&B, a converted railway station a couple of miles east/southeast of town on the coastal road toward Anstruther.

The British Golf Museum is fascinating even for non-golfers, and it has a lovely cafeteria upstairs overlooking the Old Course 1st/18th holes.

The walking tour of the Old Course consists mainly of standing around on 1st/18th holes listening to anecdotes of famous players and golf jargon, and if you're lucky your guide will give you a brief photo opportunity on the Swilcan bridge.

Playing the Himalayas has to be the best value in Scotland. Note that the kiosk where you pay and get your ball and club is around the side of the Ladies Clubhouse; the sign "Members Only" refers to entering the building, but the public are welcome to go around the side. Himalayas has no lockers or other facilities to hold your belongings while you play.

Another vote for the Himilayas golf course at St Andrews. It cost 1 pound and I was golfing 50 feet away the "pro" golfers and their caddies on the old course. So much fun and a real bargain.