They don't have their rates loaded out into September yet, but if you can swing it, try Hotel Lungarno. Their rates are loaded through July, and the end of that month is 248 euro a night. That's peak season, while mid-to-late Sept isn't, so I'd think it may be in your range at that time.

http://www.lungarnocollection.com/en/

I'm talking specifically about Hotel Lungarno, they have some other properties that will be on that site too. I can't speak for any of those, but they're expensive

I've stayed here and it's a nice place. Not over the top with amenities, but it doesn't have to be. The hotel's restaurant is actually pretty decent if you're into that - we ate there on arrival since our room wasn't ready. It's also located Oltrarno, which is on the other side of the river from the historic center, maybe a 10 minute walk to the Duomo. Oltrarno is a nice area, not quite sitting in the middle of all of the tourists, but Ponte Vecchio is right there. It's walking distance to some of my favorite places to eat and drink in Florence, many of which are on that side of the river.

I'd definitely send a couple sans children here. We were there on our honeymoon as well.