I'm not really sure you understand what the RS-232 does. It's only a transport mechanism that sends data using some format that is hopefully documented. It isn't actually related to quantization at all or in any way related to accuracy.
So what we really need to find out is how the device actually performs measurements and how it generates data output. From what I can see by reading the promotional flyer you linked, I'd assume that the device does internal analog to digital conversion at some stage and outputs that information on its display. Don't assume that its analog to digital conversion system as a whole is worse than the PC's and that it's only the number of bits on the ADC that matters. Unless explicitly stated otherwise in the manual (I sure hope that you have it or can at least obtain it!), I'd expect that the calibration and resolution/uncertainty data to relate directly to the data shown on the screen. At that point, you already have one source of quantization noise, but I wouldn't worry too much about it. It's up to manufacturer to provide proper calibration data and make sure that the initial analog to digital conversion quantization noise isn't a problem.
Next we have the output. I'd expect the device to send data which is at least as precise as shown on screen using RS-232, because it already must have that data in a digital format in order to display it in the first place.
I'd also suspect that the analog output is created using digital to analog conversion and that it is based on digital data. That suspicion arose from mention of digits in the accuracy section of the linked document. Unless it's explicitly stated in the manual that analog output is derived from initial analog signal, I'd expect it to come from digital data displayed on the screen. As I mentioned that few times before, do check that in the manual.
Also note that even if the analog output is based on actual analog source, there's no guarantee (unless it's stated explicitly in the manual) that it will be of better quality than the data sent over RS-232.
If the output is based on what I suspect, then you'd have two more sources of quantization noise: Digital to analog conversion at the output and then again analog to digital conversion at the input at the computer.
You said that the computer captures both analog data and RS-232 information. Why don't you do some checking on expected values and see which one seems to be more accurate in your use case? Also note that if you use analog output and it is set up as I suspect it is, it could be creating false precision from the extra noise which comes from the DAC, coaxial cable and ADC on the computer.
In the end, remember that you have the manufacturer of the device too! Why don't you ask them which output type is more precise? The worst that can happen is that they don't answer your question.
Short version: Find a real manual for the device and read it! It should have data you need and if it doesn't, ask the manufacturer. If you can't do that either, then don't touch is already working! Do try to understand it though.