Simple answer: Carb at home and take it to the mountains. For 2.5 volumes in the mountains set the regulator to 15.7 psi and use 5' of 3/16" tubing to serve the beer.
Long answer:
I'll have to make some assumptions but here is what I would do.
And let's make sure we're all using the same terminology.
atmospheric pressure = the pressure exerted by the weight of air above it (at sea level this equals the 14.7 psi you listed, while at 6732 ft there is less air above pushing down so the atmospheric pressure is only 11.3 psi). If I put an "a" after psi it means pounds per square foot vs atmospheric.
Gauge pressure = pressure referenced vs atmospheric pressure. a "g" after psi means this measurement is gauge pressure.
Absolute pressure = the pressure referenced against a vacuum. or atmospheric pressure + gauge pressure.
You never said exactly what level of carbonation you are going for so let's guess you are going for 2.5 volumes. If you want that level of carb at 65°F at home you would need to set your regulator to 28psi gauge (I'm gettting those pressures from
http://www.kegerators.com/carbonation-table.php). Now you let it equilibrate over those 4 days then disconnect the C02 from the keg.
If you kept a pressure gauge on your keg and dropped the temperature from 65° to 40° you could watch the pressure drop from 28psi down to 12 psi. It's the same volume of CO2 in the keg but as it gets colder there is more space between the beer molecules so the CO2 can move in there easier and there is less pressure exerted.
Next if you had the keg at 40°F at sea level, properly carbonated at 2.5 volumes, the gauge would show 12 psig which means the absolute pressure is 12psig + 14.7 psia = 26.7 PSI absolute. . If that keg is then driven up 6732 feet into the mountains (and kept at 40°) the absolute pressure in the keg is still the same at 26.7 PSI. But now the ambient pressure is 3.4 psi less. So your gauge reads 15.4 psig (26.7psi-11.3psia). This is where you would want to set your regulator to maintain 2.5 volumes of CO2 at 6732' elevation at 40°F.
It doesn't really matter when the temp change occurs. So you can carb it at home at 65 at 28psig, then drive to the mountains and set the regulator at 15.4 and let the beer cool down to 40°F.
Now we need to look at your dispensing method. Ideally you want the beer to leave the tap gently at just a smidge over 0 psig.
The equation can get more complicated but if we assume you will hold the tap at the same level as the keg we can eliminate the elevation difference between the liquid into the tubing and out of it. Then we are left with just manipulating the length and size of the beer line. A 3/16" inner diameter tubing has a pressure drop of approximately 3 psi/foot. We simply take the pressure and divide by the pressure loss per foot to find the length of tubing required. 15.4psig(3psi/foot) gives 5.23' of 3/16" beer line.
Patterns
BN Army // 13th Mountain Division 
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