For the Doubletree Sirkeci, for a 07:40 flight to Athens, leaving the hotel around 04:30-05:00 would be highly enough and taxis should be readily available at that time. Just ask for a yellow taxi from the reception and the taximeter would cost no more than ₺250 ( about € 16-18 ), plus there shouldn't be traffic in that direction.

Booking a private transfer or a pre-arranged transfer would mean that you would be paying a huge tourist mark-up for additional comfort which is IMO not worth it as a local.