Another issue here is how fast you must be able to make a reading, and how fast the current may change. Measuring a value that changes means that you must either have a sample-and-hold circuit or have a fast ADC in relation to the signal. And fast also means that it reduces your opportunities to perform multiple measurements and average them to reduce noise.

Another issue - how big steps does a range-change represent? A factor 10 as in a normal DMM, or a factor 2 or something else?

Yet another issue is that when you switch range, the steps in your data capture will be correspondingly much larger. That means that a 1-bit noise will jitter your value to a larger degree.