Single shunt with multiple current sense amplifiers vs using multiple shunts
The simplest approach is a single shunt with a single switchable gain amplifier. Whether you can use this depends on your precise circumstances.
What is the largest shunt resistor you can use at maximum current? This will be limited by your voltage drop. For instance, at 150 mA, a 1 Ω resistor will drop 150 mV. Maybe you can tolerate more?
Now with that shunt resistor, can you specify an amplifier that will give you sufficient accuracy (noise, offsets) at your lowest current? 1% accuracy of 1 uA is 10 nA, or 10 nV with the same shunt resistor.
There are some good autozero amplifiers available now, but they will not get you to a confident 10 nV input resolution, you might see offsets in the μV region. As you head into the nV, errors appear from everywhere, Seebeck effects from thermal gradients on the board for instance. Maybe you don't need 1% accuracy at the lowest current range. Maybe you can reduce the bandwidth with heavy filtering to improve resolution at sleep currents.
I see little point in using multiple amplifiers, expect for the dubious advantage of not having to switch their gain. Better to spend your money on one really good amplifier and switch its gain.
If you can't handle the low current with a single shunt resistor, then you need at least two shunt resistors, whether switched by some form of FET switch, or automatically routed by diodes. Note that using diodes will give you a large variable voltage drop as the load current changes. If you can tolerate that, then can you tolerate a single, much larger, shunt?
I am a great fan of using a silicon diode, or better still a diode-connected power transistor (as they track a logarithmic law more faithfully), for this sort of very wide range current measurement. However, your 1% accuracy requirement is completely beyond using a diode. Now if you could be happy with 10% accuracy, then it gets to be possible.