So you are saying that you don't care about signal quality, because you think LabVIEW can filter away the noise? When filtering away noise, you will also filter away signal data. Remember that it isn't DC signals you are sampling - it's the AC part of the signal you are interested in. Saving a dollar on filtering components may cost you thousands of dollars trying to filter the signal, and still fail.

It is only under some circumstances that you can filter and extract signal data that are weaker than the noise. That is something you do, when you have no alternative. And there is a cost involved. If the signals are repeating identically, you can't combine multiple curves to reduce the noise influence. The ECG signals may be mostly repeating, but any one-off anomalies will be affected by the noise suppression, when the software stacks and averages the curve from many heart beats.

Anyway, with ECG, the scaling factor shouldn't be important, so it is enough that the voltage reference has a stable voltage, even if the voltage may be off by a number of percent. I would be very surprised if 10 bits would not be enough. With a correct design, I would think 8 bits would do fine. Most digital oscilloscopes only has 8 bits of vertical resolution, and that tends to be enough with the normally used size of monitor. But as already mentioned - bits are not meaningful unless you design the hardware so that you measure good data.