  We explore a reduced phase space quantization of loop quantum cosmology (LQC)
for a spatially flat FLRW universe filled with reference fields and an inflaton
field in a Starobinsky inflationary potential. We consider three separate cases
in which the reference fields are taken to be the Gaussian dust, the
Brown-Kucha\v{r} dust, and massless Klein-Gordon scalar reference fields
respectively. This is a "two-fluid" model in which reference fields act as
global clocks providing a physical time in an inflationary spacetime, and allow
bypassing various technical hurdles in conventional quantum cosmological
models. The reduced phase space is obtained in terms of the Dirac observables
of the gravitational as well as the inflaton degrees of freedom. The physical
Hamiltonians of the two dust models take the same form but turn out to be quite
different from that of the Klein-Gordon reference field which reflects an
aspect of the multiple choice problem of time. Loop quantization is implemented
using the so-called $\bar \mu$ scheme and the Schr\"{o}dinger equations
involving the physical Hamiltonian operators generating the evolution in the
physical time in the dust and massless Klein-Gordon models are obtained. These
turn out to be quantum difference equations with same non-singular structure as
for other models in LQC. We study some phenomenological implications of the
quantization using the effective dynamics resulting from the reduced phase
space quantization including the resolution of the big bang singularity via a
quantum bounce, and effects of the different reference fields on e-foldings in
both the pre-inflationary and the slow-roll inflationary phases. We find that
different clocks, even when starting with a small but same energy density, can
leave tiny but different imprints on the inflationary dynamics. (abridged)
