Some great points above, especially thinking about wind resistance. I ride a DRZ400E, (and a 1200GSA) and while it's not great on the highway it's ok. Many eons ago, I rode a 250 Ducati Scrambler all over BC, and it was certainly slower, and FAR less reliable than your KLX. I know a LOT of people who have done big trips on small bikes. A 125 scooter across the USA comes to mind, a step-thru Honda 90 top to bottom of Africa and many others. There's a BC couple who rode to Ushuaia on a pair of KLX250's and thought they were perfect.
My tips would be similar to above, streamlining and weight MATTER on a small bike. Tank bag and tank panniers can really help with keeping the load in tight.
Also, don't ride at night, you're small and easily not noticed - but having said that, put some reflective stuff on the back, and as wide as you can - and this is a tip for everyone - a good reflective sticker on the back of your helmet is a HUGE improvement in visibility - all of a sudden you're a lot bigger and closer to someone following.
Finally - don't sweat it, just go and have a great ride!