Earth scientist here. Unfortunately, although you're 100% right, we can't use C-14 as an isotope fingerprint for human caused climate change because the data is tainted by atomic bomb testing in the 50s and 60s.
Until now. Only in the last few years have the atomic bomb created C-14 isotopes in the atmosphere come down enough that we can finally start using it to determine carbon sources.
A much better one to use is C-13. Carbon-13 is a stable isotope just like standard Carbon-12. But because it's slightly heavier it's more difficult for plants to use it as a source for their carbon capturing to make sugars. It's like stopping your car, it's slightly harder to stop your car if you have a passenger then if you're alone. If you're trying to manufacture sugars at the best possible energy level you need as much C-12 as you can get.
The ocean doesn't give a shit. Rocks (that use carbon such as limestone) don't give a shit. Volcanoes don't give a shit. Only plants have trouble sequestering C-13. So C-13 levels in plants are lower than they are in other reservoirs of Carbon: the ocean, the atmosphere, rock, and magma (volcanism). We know this because we can shove them into mass spectrometers.
So if you want to find a source of atmospheric carbon follow the C-13. If it's the same from day to day and year to year you know your carbon source is one of the following: the ocean, volcanoes, or rock.
If C-13 in the atmosphere is falling, despite CO2 rising. That tells you the source can only be plants, which is what coal and oil are made of, ancient plants.
And guess what?http://cdiac.ornl.gov/trends/co2/modern_isotopes.html
The argument is good and settled. Humans burning fossil fuels is causing climate change.