I'd definitely recommend making it a two-day hike and doing both sections. What's nice is that if you want to pay out a little bit of money (20-something these days?) you can stay at Carter Hut. Don't carry a tent, stove, or filter, just bring food and get the run of the kitchen, plenty of water, and a bunk to sleep in. That will take quite a bit of weight out of your pack.
If you'd rather avoid the hut, with a hammock I'd recommend looking south of Carter Notch on the Wildcat River Trail, just outside the forest protection area, unless the weather is warm and calm. The ideal spot, up on the height-of-land of the Notch, can get pretty windy.
Either way, fill up with potable water at the hut. There's no water up on Wildcat Ridge, and no water once you pass the spring on the side of the Notch going up the Dome until you get to Zeta Pass, and after that I remember only a little bit near the Pass until all the way over North Carter.
There are plenty of hikers who will do the whole thing in a day (ie, go from Pinkham all the way to Imp). It's doable for a really strong hiker. And if you were day tripping, hiking out instead of going to Imp, you could carry a lot less. But if you choose to do both ranges ina day, you have to keep moving through spots where you should really sit down and enjoy the view and the moment.