Personally, I'd put some new tires on. If you choose to do so, don't put them on right before your trip. Give yourself some time/miles to make sure you like them and to note anything weird going on with the tire/tube or related to the installation.
Lots of choices for spare tubes. For some unknown and most likely pointless reason, I use UHD tubes in my tires. I carry a regular-style tube in case I have to change one roadside. The standard tubes are a lot less bulky and easier to work with, especially getting a new one in.
As far as brands, I have no recommendation. I would think any tube from a reputable manufacturer would do the trick for a spare or for running in your tire all the time.
I'm sure there are folks here who know a lot more about tires than I do and I may be wrong, but the remaining tread you describe on your stock Dunlops sounds a little scarce. If you have the means to do so, why not just put on new tires/tubes so you don't have to worry about it throughout your trip?
Standard Obligatory Quote: Original Source Unknown:
"90% of motorcycle flats occur in the last 10% of a tire's life."
Also, depending on when you got your '09, those tires could be at least two years old.
They say the age of a tire is just as important a consideration as tread depth when it comes to serviceability, but I'm not sure how "old" a tire has to be in order to be considered "compromised."
Enjoy your trip! Sounds like a fun one.