The R3 with it's 321cc engine will hold up during interstate trips if doing the trip solo. A new, well maintained 300cc motorcycle can be ridden across the United States, easily and reliably. If you plan on carrying a passenger, maybe a 500cc or bigger, would be better.
I have taken my R3 into other states, from my state of Pennsylvania to West Virginia for instance. I have ridden the R3 on the highway 8-12 hours during a day. The R3 is more than capable of handling long road trips like the ones you are talking about. A better question would be, are you capable of riding the R3 that kind of distance?
My biggest issue with highway riding is the wind. Small windscreen and too much wind noise. Take a day trip on it. Make sure to spend 8-10 hours on the highway running the bike between 60-80 mph. Then, you will know.