We went to both Split and Dubrovnik last year (loved them both!). I wouldn't really have picked up on GoT references if I hadn't been with DH (he watches, I don't...) I don't think you need to do a GoT tour - just look up some of the filming locations before you go, and have fun finding them yourselves. I think they centre on the palace anyway from what DH said when we were there.
I preferred Dubrovnik, but if you can't get there, then Split is a very good choice.
For the amount of time you've got there, I wouldn't go to one of the islands, or to the Nat Parks - you'll just spend too much time travelling. I'd do the following...
Tues afternoon - explore the palace. Wander around the old town, find your bearings/enjoy getting lost etc.
Weds (or Thurs) - Marjam national park - if you like to walk, then you can lose yourselves here. Go to the beaches, spend time at beach bars, wander along the coast and through the forest. Easily a nice chilled day.
Thurs (or Weds) - go to Trogir (either by water taxi or bus). It's got more of a similar vibe to some of the towns on the islands. Just spend the day wandering around the old town, eating and drinking, going up a tower or 2 for the views.
Stay in the old town, for sure - you're better connected for transport, and there are loads of eating/drinking options. A big/chain hotel with pool is wasted when you're doing a city break and you want to be out and about.