Just a quick follow-up. I ended up staying in Ramada Üsküdar and the location worked out well for me.
Late night on Friday (touch down around midnight then a ~25mins walk on the travelators to immigration at the new airport) the taxi took ~45mins and cost 160TL.
The hotel was right next to a metro station - I took that in the morning to Üsküdar ferry station and ate breakfast overlooking the Bosporus. I then took a ferry to Eminönü and walked around Sultanahmet. I took a train+metro back from Sirkeci at 4pm, for a taxi at 5pm (based on the hotel's advice) which got me to SAW at 17:45.
If anyone has a similar stopover situation in the future (swapping airports between IST & SAW and staying over for a night and a day) and wants to be close to central Istanbul I can recommend Ramada Üsküdar.