I live in Skye. If you are spending a few days in Edinburgh before coming to Skye then I would get the tram back out to airport and rent a car there before driving to Skye. It will take you about 6 hours to drive without stopping, but it would make a very nice day trip with perhaps lunch in Pitlochry or Dunkeld on the way. You could drive to Skye via the A9, then A86, then A87. You could return via the ferry to Mallaig, then A82 to Fort William & Glencoe and turn east on to A85 at Crianlarich before reaching the M90 at Perth, which will take you back to Edinburgh airport to return your car.
If you take the train to Inverness and rent there you will probably have fewer rental options in the town centre than at the airport. The drive from Inverness to the Skye bridge without stopping is about 2.5 hours.
There is only one rental company on Skye - Morrison. They are basically a car repair garage who used to just rent cars to locals when their own cars were off the road. They have branched out a bit more recently, but probably wouldn't have an automatic shift car if that's important to you. They are based in Portree so you'd have to take get from Kyle of Lochalsh to Portree by bus. Although they are a great little company I wouldn't generally recommend them for tourists. There is also Morar Motors in Kyle of Lochalsh - another small company.
Whatever car you rent I would strongly advise you to avoid getting an electric car. Charging points are few and far between here and often out of service.
You say 'this year' which is ringing slight alarm bells in my head as depending when you plan to come you might find it hard to find good accommodation.
Skyegirl (Jacqui)