Googling around for ways to measure the viscosities of shear thinning liquids, it seems to me that most of the time viscometers are used at different settings to measure different apparent viscosities. Is this the only strategy out there, did I miss something?
The sort of rheometers normally used for industrial research work by applying a controlled stress then measuring the resulting shear rate. Typically you program them to start at a high stress and end at a low stress, and the rheometer will go away and automatically take readings at intervals between the two. When it's finished it spits out a table of stress-strain readings. The software running the rheometer can display this in any way you want e.g. as a graph of viscosity vs shear rate.
You can put whatever type of fluid you want in your rheometer. You just need to bear in mind that with strongly shear thinning fluids it can take a very long time to obtain the low stress readings because the rheometer has to wait a long time for the reading to stabilise. You also generally need a trial run to get a rough guide as to what initial and final stress to specify.