Your information is significantly out of date. The current percentage of CO2 in the atmosphere is about 0.0427%, not 0.03% as you claim - it's been rising since the start of the Industrial Revolution, when it was about 0.028%; It hasn't been as low as +/- 0.03% since the 1950's. While there is a certain (small) amount of debate about how much CO2 was in the atmosphere during the Pleistocene era (2.58 million years ago up to about 11,000 years ago) because the proxy data that we have are not as reliable as modern instrumentation, we have pretty high confidence about the data gathered during the 20th and 21st centuries, and they show a consistent rise in CO2 levels that really started to take off around 1950 or so. Moreover the source of the bulk of this excess CO2 is clearly from fossil fuels, which can be demonstrated by the atmospheric carbon and oxygen isotope levels which precisely match what would be expected if the source of the excess CO2 was all ancient plant matter, and no other explanation (volcanism, etc) seems to fit the evidence. The quantities required also roughly match the amount of fossil fuels burned (there will of course also be some released from melting permafrost, which will have a similar isotope profile, but so far that's a secondary effect, and is in any event a knock-on effect).
This of course doesn't *prove* that the additional carbon is *causing* climate change, and the computer models are, like all computer programs, subject to bugs and incorrect assumptions and the like. There is no possible way to run multiple experiments on additional (real, non-simulated) Earths to see what effect different levels of CO2 have.
HOWEVER, we do have pretty unequivocal evidence that the global climate is warming up significantly. Probably the most obvious is the loss of ice from the poles, especially the Arctic. For a while it was thought that the same phenomenon was not happening in the Antarctic, and that there was a net increase of ice in some parts of the Antarctic, but that no longer appears to be the case. It is hardly a requirement that temperatures and effects increase by equal amounts all over the globe (!). Likewise the glaciers in high altitudes are almost universally retreating.
Another piece of evidence that is hard to explain any other way is that the global sea level is moving inexorably higher over the last 200 years. Not just at a few locations (which might be caused by other effects such as subsidence), but all over the globe. This is mostly because of the thermal expansion of the water in the ocean, not the melting of polar ice since so far most of that has been sea ice rather than land ice, and melting sea ice won't raise the water level. It's not very fast (less than 2.5 mm/year on average), but it is extremely consistent, and over a span of decades can put many low-lying coastal communities at risk.
And of course we have direct temperature measurements taken at thousands of locations around the Earth, though I'm sure you will want to discount them. While there is every effort made to ensure that the measurements are a good representation of average global temperatures, it is never possible to have one thermometer every square kilometer so that we can get a complete global picture, and there's always a small chance that the dataset is not fully representative. They too show a consistent rise in global temperatures.
Which brings us back to those pesky computer models. While there are always possibilities for errors in the models, the awkward fact is that we have several independent models, and they all show more or less the same thing: That the higher levels of CO2 are largely responsible for the observed warming effects. And when you run them on PAST levels of CO2, their predictions all agree fairly well with the historical data. It is difficult to explain why they would all be wrong on future predictions (and in the same direction) when they all work so well for the historical climate data. It may not be quite impossible, but it certainly strains credulity.
It was possible to have some reservations about the accuracy of the climate models 20 years ago or so, but it is getting harder every year to deny that their predictions should be taken seriously.