Private tours in Iceland

We have never done a private tour as we self drive, but we have used Troll for guided day excursions and they do private tours. We have always had great guides and been very pleased with Troll in general, FWIW.

Here is a link to a private tour of the Golden Circle that they do. This tour is 12 hours, but since it is private you should be able to make it shorter if you want.
https://troll.is/day-tours-iceland/day-tours-from-reykjavik/golden-circle-farm-visit-private/

I have not used Nice Travel, but friends have and were pleased. Here is a link for their 8 hour private tour.
https://nicetravel.is/iceland-tours/classic-golden-circle-and-kerid-crater-private-tour/

I know there are other companies, but these are the two that I know the most about.

Based on your prior posts, it will be necessary to verify a private tour can take your young children, as they'll need boosters seats in any vehicle.

FWIW I took a GeoIceland small group tour to the South Coast and they also do private tours. It is an extremely long day (11 hours) and I think that would be a drag for your kids. Your Friend in Reykjavik (I used them for a food tour) has a listing of family friendly tours: https://yourfriendinreykjavik.com/family-friendly-tours-in-iceland/

I have also used Troll and we really enjoyed their Golden Circle tour. My husband has mobility issues and I emailed them with questions regarding their various tours and his participation and they were prompt with replies and suggestions. I cannot comment on the other companies. We did go with a group however, you would want to contact them for private tour information