While normally bit depth would be associated with a more accurate signal to avoid quantization noise, I am wondering if recording at a higher sample rate will allow a more accurate interpolation of digital volume increases in post processing to avoid reduce this noise.
Short answer: no. At the limit of converter precision, the error is not correlated well enough with the residual signal to provide reliable dithering. That only makes sense when using converters below their available accuracy (why would you do that?). When reducing precision by some sigma-delta kind of conversion not at the limit of its accuracy, higher rates may indeed recover relevant data through low-pass filtering an oversampled data set
The yield is quite less than using the conversion circuitry at its nominal precision, however. So there isn't really much to be gained unless the data depth has been reduced before transmission by some process that you have some insight into and the actual precision to which the data is correlated with the available readings is higher than the nominal precision.