The Wildcat River Trail was not broken out a week ago when I was at Carter. In fact, you'd have to break deep trail just to get to the Rampart. I don't know if it's changed since then. However, if you're looking just to have some fun, this may be fantastic. Use caution out on the Rampart. The views are well worth it, but wear snowshoes and use caution when stepping from boulder to boulder - you don't want to fall between them. I've done it in winter, it's do-able, just be careful.
If you take 19-Mile to the Carter Dome Trail and stay on that trail all the way to Carter Dome, bypassing Mt. Hight, and come back the same way, you can do it all with snowshoes. Mt. Hight, however, has the best views in the range and I'd recommend bringing crampons and going over it. Regardless of the route, watch out for very deep drifts in the col between Hight and the Dome ... spruce traps abound.
If you're going up from the Notch to the Dome, the first 1/4 mile is very steep and you'll need crampons. After that, however, it's nice and moderate and not difficult.