It's difficult. Engineering is difficult. If it was easy, everybody would be doing it, and engineers would be paid less. You have to do the work.
Start from an accurate specification of what's needed. The specification might require accuracy over a period of time, or a temperature range, or linearity over a range, or good tracking between the 1x and the 1000x ranges. Does the use case require continuous measurement, or are there pauses in operation? Each of these requirements might push you in one direction or the other.
Note that 'as good as possible' is not a specification. You need to know what your customer wants. Many times I've had to ask the question of a client 'do you really need this spec point here, because if you do, it's going to double the cost?'
General advice. Somewhere, a precision instrument has to go back to a precision reference. Generally, one precision thing is easier to maintain than several. Passive precision (resistor ratios) is easier to get right than active things (voltage references). Time can be made orders of magnitude more precise than voltages and currents (crystal oscillators, PWM, delta sigma).
You can use these resources to calibrate less accurate measurement channels. However, these channels need to be stable (over time, temperature). If you need more expensive components for stability, do you get precision for free at that price?
How long will it take to calibrate these channels? How often do you need to calibrate to compensate for drift? Do you have to interrupt the measurement to calibrate, or can it be done in the background? Do you need two measurement channels and ping-pong them with calibration (like auto-zero opamps do)? Note that calibrating on switch-on is the worst possible time to do it, as the instrument will be used warm, so calibrated at the wrong temperature.
You'll probably find yourself pushed in the direction of having a one high quality voltage reference, one high quality time reference, one high quality resistor to get between volts and current, and a set of ratio resistors to translate accuracy between ranges. However, you'll need low resistance low leakage switches to juggle these resources around, so you may be better off having dedicated precision in multiple places, or use relays in key places instead of CMOS switches.
I don't know about your precise situation. You won't know until you've done quite a lot of detailed work. That work should include prototyping some measurement channels and measuring how they perform. You
might will learn something unexpected.