I won't comment on the double RtoR Grand Canyon hike other than to say it will probably be very snowy at the North Rim. With roads to the North Rim closed, you can't bail out there, should something go wrong. I would think that down and back from the South Rim would be a better choice. There are connecting trails in the canyon for possible loops, so you'll see a lot more of the Canyon. Check the nps.gov website for details on trails, permits, etc.
However, there's no public transport to the GC (check the park's website), and the park shuttle busses don't run in winter. You could get a charter flight to the Grand Canyon Airport and then a taxi. If you can somehow get to Williams (does Greyhound stop there?), there is a train to the GC but it's a tourist excursion type thing, quite expensive. You might be able to persuade one of the bus tours from Las Vegas to let you make a one-way trip each way; you'd have to check with the individual tour company to see if this is possible for folks with backpacking gear. (A lot of tours won't take hikers because they "detract" from the tour experience--an example being Gray Line tours to Mt. Rainier.) Renting a car is probably your cheapest option!
The combination of being accessible by public transport, suitable for a 4-5 day hike and snow-free in the dead of winter eliminates about 99.9% of the possibilities. The first criterion, public transport, is by far the most restrictive. Rent a car, and there are many, many possibilities, especially in the Southwest.