About 10 years ago I did a 2,600 mile summer trip on my newgen. Your assessment is pretty much right... it can certainly do it, but acceleration is limited. The 650 is a good transportation appliance and not a bad idea. It is in many ways like a big Ninjette.
Given your height you'll likely find the 250 quite cramped after several hours. I'm 5'7" on a good day so I fit the Ninjette better.
But I wouldn't dismiss the 250 out of hand. It's actually kind of fun to talk to people who think it's impossible to take a trip on a little bike.
I did my journey using an Axio hard-shell tank bag and an SW-MOTECH Cargobag universal tail bag (does not require mounts). This worked well and carried everything I needed for the trip.
I'd go with a big tank bag. Not only is this handy, but on really long stretches you can rest your chest on it and take some weight off your hands.