  We investigate the impact of a mean field model for the $\alpha\Omega$-dynamo
potentially active in the post-merger phase of a binary neutron star
coalescence. We do so by deriving equations for ideal general relativistic
magnetohydrodynamics (GRMHD) with an additional $\alpha-$term, which closely
resemble their Newtonian counterpart, but remain compatible with standard
numerical relativity simulations. We propose a heuristic dynamo closure
relation for the magnetorotational instability-driven turbulent dynamo in the
outer layers of a differentially rotating magnetar remnant and its accretion
disk. As a first demonstration, we apply this framework to the early stages of
post-merger evolution ($\lesssim 50\, \rm ms$). We demonstrate that depending
on the efficacy of the dynamo action, magnetically driven outflows can be
present with their amount of baryon loading correlating with the magnetic field
amplification. These outflows can also contain precursor flaring episodes
before settling into a quasi-steady state. For the dynamo parameters explored
in this work, we observe electromagnetic energy fluxes of up to $10^{50}\, \rm
erg/s$, although larger amplification parameters will likely lead to stronger
fluxes. Our results are consistent with the expectation that substantial dynamo
amplification (either during or after the merger) may be necessary for
neutron-star remnants to power short gamma-ray bursts or precursors thereof.
