Definitely the 13 day tour - you get to see so much more of Scotland.
I'm surprised that people tend to avoid June and July. June is one of my favourite times to visit Scotland. Weather tends to be good, the wild flowers are in flower and you get very long days too.
Taking on extra time either before or after the tour - chose either Dumfries and Galloway OR Borders and Northumberland. These are areas not covered on the tour and are also places not on the usual tourist tick list BUT popular with 'those in the know'. You could easily spend a week in either of these places. Have a look at Undiscovered Scotland for ideas (and lots of pictures)
Dumfries and Galloway has some superb scenery, worthy of the Scottish Highlands around Glen Trool. It also has the Mull of Galloway with Logan Botanic Garden (one of the best gardens in Scotland) or the Logan fish pond (go and hand feed the fish!) There are the ruined abbeys of Glenluce, Sweatheart and Dundrennan. There are the ruined castles of Cardonness, Carsluith, MacClelland's Tower. There are the prehistoric Cairnholy Cairns. There is the early Christian heritage of Whithorn and Wigtown which markets itself as a Book Town. The Gem Rock Museum in Creetoiwn is wonderful and worth finding. There are small villages to explore - Castle Douglas, Kirkcudbright, Gatehouse of Fleet. ...
The Borders have the great ruined Abbeys of Melrose, Dryburgh and Jedburgh as well as some of the best stately homes in Scotland (Traquair House, Floors Castle, Manderston, Mellerstein ...) There are ruined castles at Dirleton and Tantallon. The coast is worth exploring too with places like Dunbar and North Berwick.
Northumberland has the coast, Cheviot and Hadrian's Wall. You can visit the Holy Island of Lindisfarne, reached by tidal causeway, or do a boat trip to the Farne Islands., There is Bamburgh Castle and also Alnwick Castle with its stunning modern garden. There are the ruined castles of Dunstanburgh or Warkworth. Alternatively there is Cragside, the C19th house of a Victorian industrialist and inventor and often described as the 'house where modern living began....'