Hi Jason,
Not sure if you can directly "read" the value of the chemical potential from NMR data (e.g. T1 relaxation time), but from data shown in this paper by Weber and Kimmich it is clear that T1 relaxation time will respond to changes in the chemical potential of a polymer.
Fig 1C. shows plots of T1 vs larmor frequency (strength of magnetic field) at different concentrations of a polymer. Since concentration is a "component" of chemical potential quantity there very well may be some correlation. Still, I think it would be worthwhile to run a series of control experiments where you know the μ (chemical potential) and can measure T1, to prove the concept and maybe "calibrate" your measurement.
The data obtained in the paper used a specially built relaxometer where strength of the magnet could be adjusted. If you only have access to a standard NMR instrument with a fixed field - you'll benefit most from a low-field instrument as changes in T1 are more sensitive to various sample parameters at lower field.