Itinerary Advice

One advantage of Inverness is that there is plenty of choice for hiring a car. Arnold Clarke (or Celtic Motors which is the US agent) comes very highly recommended and has a base in Inverness. The rail journey is scenic and there's only one route...

You will need a car to visit Ardverike and also on Skye. Visit Glencoe on the way back from Skye.

When are you planning the trip as the puffins will be beginning to leave in August.

The trip is planned for April. An alternate idea to Inverness is Glasgow.

My ears pricked up on reading that you were visiting Ardverikie, as I spent close on two years archivally cataloguing the English records of the estate (they relate to Muncaster Castle in the Lake District). And it got deeply under my skin, just as it did for the Ramsden family, who had a very strong attachment to the place.
It was then a truly vast estate, far, far bigger than it is now.
They were extremely philanthropic to the local community, so it is very fitting that the proceeds of the open day on 1 June are to go to Laggan Parish Church.

The Ramsdens usually used what is now the Caledonian Sleeper to go up to Ardverikie, normally to Dalwhinnie Station, but infrequently to Tulloch Station.

Dalwhinnie Station by the way served a small part in WW2 as General Montgomery finalised his plans for D Day in a railway carriage stabled at the station. There is a plaque at the station to commemorate that.
You could stay at the Dalwhinnie Hotel.
One of the names on the Dalwhinnie War Memorial was the son of the owner of the Hotel. For over 70 years the memorial was opposite the hotel, so that his mother could greet her lost son every morning. It wasn't relocated until the 90's, long after both parents had died.

However Dalwhinnie is a pretty small village. Newtonmore or Kingussie would be better places to stay. Personally I don't see the need to go to Inverness.
The logical way to do this would be car rental all the way.
But as you want train travel I would suggest taking the train to Fort William. Then renting a car from Fort William Car Hire (a local company).
Then drive up the A86 through Glen Spean and along Loch Laggan to Dalwhinnie etc.
On Day 5 take the A86 and the A830 to Mallaig for the ferry to Skye.
On Day 10 come over the bridge off Skye, and overnight at Fort William.
On Day 11 Day Trip Glencoe, turn the car in at FW, evening bus to Oban.
On Day 12 the Puffin boat trip from Oban (holding yourself hostage to the weather co-operating).
On Day 13 Train along the Oban branch, then back to Glasgow and Edinburgh. Or there is a direct bus #976 at 0920 from Oban to EDI Airport arrive 1300, via the Trossachs. (it is a seasonal bus so you won't find it online currently)

You could do the Falkirk Wheel from Edinburgh on Day 2.

That is the only way I can see you fitting in a scenic train ride.

PS- If this is April you need to be booking somewhere on Skye last week or last month. As this is April the #976 bus above won't work- it starts the season in late May.

Thank you for your time and detailed response. I may forego the scenic train and Inverness. I do have lodging on Skye.