Well, I solved my problem (thanks for your replies) and thought I'd come back with a brief review of the property.
Overall, a good stay. A couple of small complaints though. Firstly, the (standard king) room really was quite small. Secondly, the bathroom had a weird moldy smell -- I'm sure I could have got a different room if I'd complained, I didn't care that much, it was just a bit off. Thirdly, on one night the maids managed to take away the towels but not replace them, leaving me with no towels! Luckily they didn't take the bathmat so I dried myself with that. Still, it's a bit of a slip-up.
Best part: the breakfast buffet. Huge selection of food, with the highlight being the hanging honeycomb which you can scrape onto your bread with a knife. As a lowly Gold (thanks to a status match off another status match, thus totally undeserved) I got a fistful of breakfast vouchers plus a welcome platter of fruit and Turkish delight, plus two bottles of water a day, plus a goodbye platter of delicious cake, which is the best treatment I've seen for a non-top level status guest anywhere.
On the subject of airport transfers: I'm a cheapass and don't feel like dealing with the possibility of getting ripped off, so I just took public transport -- surprisingly good and fast, but often very crowded. The tram stops right outside the hotel and connects with the train to the airport. I think the whole journey takes about 40 minutes and costs... I dunno, a few bucks.
Overall it was good, but if I did Istanbul again I'd probably stay on the other side of the Golden Horn. The old part of town contains very little apart from tourist attractions, tourists, and people who prey on tourists. Of course you'll want to see all these things, but you can see them during the day and retreat back to the new town at night to find places for dinner that don't suck.