The other thing I'd say is dont underestimate how long each site will take. Topkapi Palace for example is probably 3/4 to a whole day in itself. I see the Istanbul Explorer web site suggest 2 hours, which is ridiculous.
The number of significant site is fairly limited - Topkapi Palace, Dolmabahçe Palace, Galata Tower are the main ones. They offer a Blue Mosque guided tour, but visiting the mosque is free. The Bosphorus cruise you could just take a (cheap) ferry... You won't have time to go to Bursa, but again you could just yake the ferry (which is.more fun than driving on the tour they offer, which strangely skips the Bursa Grand Mosque).
So i personally doubt you'd save money, and you'd lock yourself in and not allow the reedom to just explore.
Lastly, one of the key sites in Istanbul are the mosques, and (apart from the Hagia Sofia) they are free, unless you want to make a donation. We visited 11 over 10 days - my three favourites were the Arap Mosque (originally a mosque, converted to a church and back to a mosque), the Mirimah Sultan Mosque (in Üsküdar on the Asian side) and the Fatih Mosque, which dates back to the 1700s and is enormous.