Pre or post Scotland tour

Fly into Edinburgh pre tour and spend some extra days there. The tour is only in Edinburgh for a very limited amount of time and there is a lot to see and do there.

think twice about the Scottish 500

https://www.thescottishsun.co.uk/travel/13267374/nc500-investigation-tourists-locals-route-reality/#:~:text=THE%20North%20Coast%20500%2C%20dubbed,8

Skye might be worth a visit if you are interested in historic geology. For 3 Billion years the crust of NW Scotland was joined to Greenland and the Canadian Shield. In the past 60 Million years the Atlantic Ocean has opened gradualy unzipping from South to North separating NW Europe away from North America. There is some unique geography and rock minerals in Skye that may illustrate this. A friend of mine took some fantastic photos of the Milky Way in Skye because of the Dark Sky conditions. It would only be worth it if you could stay overnight. Bon Voyage!

Thank you for your ideas and advice.

There's a lot more to see in and around Stirling and Falkirk. https://www.ricksteves.com/europe/scotland Look at "At A Glance"

You don't have to go all the way to the Isle of Skye to get dark sky conditions.

As examples you can get those in the Galloway Forest in South West Scotland, or Kielder (England, but just over the Border from the Scottish Borders).
Both of those are within two hours or so of Edinburgh. Also, unlike Skye, you are not struggling to find accommodation (or restaurants) or paying the resultant very high prices.

Also the 'Scottish 500' is actually called the North Coast 500. If you just read the 'Scottish Sun' without knowing the actual facts you might jump to simplistic conclusions. As always the truth is much more complex. Due to pressure of traffic parts of the NC500 are not really ideal in high summer. But travel in the shoulder seasons and it is not half as bad.
Many of the issues are confined to a minority of less than responsible camper van drivers. Even if you rent a camper van and travel in a responsible way and with thought for the size of the roads and other traffic then it is much less of a problem. If you are in a car and staying in hotels/B and B's it is even less of an issue.
The big problem in my view is camper van drivers who are not used to driving a relatively large vehicle on single track roads (and/or to driving on our side of the road) and don't park/camp in a responsible manner.

It is quite a delicate and complicated question.

How long are you planning for your must sees before or after the 13 day tour? That may well affect to decision of where to go.

We were going to stay 3-4 days after our tour and I was wondering if visiting isle of Skye is a good option
Thank you

Depending when your tour is you may or may not get dark skies in Skye. It really doesn't get properly dark here from May to the end of August. We are only now starting to get our dark skies back and with them the occasional glimpse of the aurora!