Is the new battery an AGM? Was the old battery an AGM? They require higher voltage to fully charge, but it shouldn't be an issue in a brand new battery. It becomes an issue after several months of not getting a full charge. A standard "Wet Cell" battery should be charging fine with 13.5v to 14.0v.
The GM diode is an alternative but takes a fuse out of the equation and in an extremely unlikely circumstance it could start a fire. There are fused options, but you have probably already left for your trip, so it won't do you much good now.
Sounds like you might have a parasitic draw or your starter might be having trouble, judging by your volt readings. You could have dirty connections at the starter. A battery voltage of 12.78 at rest for 6 hours or more should be plenty of juice to start your truck. Your lower voltage readings with all accessories on look to be in an acceptable range, but under load while moving they could be dropping lower. I have an Ultima in my 5th gen and it's been doing pretty good, but your right, you never know with remans.
Here is a video I did involving the "Dirty Parts Voltage Booster". It does the same thing the GM diode does, but has a fuse inline for circuit protection. If you have a wet cell battery, the booster might over charge it and cause problems that way. There are a few variables involved with this so more research might be needed. A quality lithium jump booster is a good thing to have as well. Hope it's something simple and your trip goes well with no troubles. Good Luck!