John,
A lot of this depends on the intensity level at which you are working.
At a high enough level, a diabetic can go with zero basal for several hours and keep eating glucose without needing any insulin for it. It basically becomes an insulin:carb ratio of 1:infinity (see this thread on FUD for a description, and also another reference to infinity here).
Your intensity level will vary depending on the grade, and overall a lot of this depends on your speed and pack weight. But at the duration you are talking about, you will most likely be a little below that “infinity” level where you no longer need any insulin, because sustaining that infinity level for 16 hours would be very difficult.
Test this out!
It will be necessary to test this FIRST, but I think something along the lines of cutting your Levemir down drastically, like 25-40% of your current dose, would be appropriate.
If you take too much basal, you will need to feed it the entire time, and that will become difficult, because the hypo’s will sap your strength.
Also, if you cut your Levemir down that much, it would only last around 6 hours, so you would need to take your Levemir a few times during the hike.
If you can get some diluent, then you could treat any BG rises with the diluted insulin. I think having the ability to micro-dose would be very helpful for you. All it takes is a vial of diluent so you can pre-mix a vial of diluted insulin, and some syringes.
Is it possible for you to get diluent? (See this link how-to-get-free-diluent-from-lilly).
If you can’t get diluent turned around that quickly, I can hook you up for no charge other than shipping.
(If all of that 
dilution stuff seems like it would be too difficult, than just make sure you have a ridiculous amount of carbs with you, so that IF you need to treat a BG rise with a 1/2 unit you will be able to “eat away” most of it.)