Give me a sec, I'm still laughing about the keg getting heavier
Whew....Ok, let's see...The end game is to have the carbonation pressure match up with the temperature you'll be serving the beer and the desired carbonation level - AND have the serving pressure match that carbonation pressure.
You can certainly carb the beer at 65°F, and there's no harm in setting your regulator so the carbonation level matches the style and/or personal preference. For a typical ~2.5 volumes, you'd want to bump the pressure up to the ~28 psi shown in the chart.
Then, once you drop the temperature down to - say - 40°F, you'd also drop the CO2 pressure down to ~12 psi, which will maintain that 2.5 volumes of CO2 throughout the life of the keg. The key here is your serving system has to be able to handle a 2.5 volume brew without excessive gas breakout which would lead to foamy pours.
It's usually that last part that bites new keggers, as they almost always start with beer lines that are too short. A good start at avoiding such issues is to use 10' of 3/16" ID beer line between keg and faucet...
ps: If you can actually detect the weight of a few cubic feet of CO2, you are one amazing person