  Galaxy bispectrum is a promising probe of inflationary physics in the early
universe as a measure of primordial non-Gaussianity (PNG), whereas its
signal-to-noise ratio is significantly affected by the mode coupling due to
non-linear gravitational growth. In this paper, we examine the standard
reconstruction method of linear cosmic mass density fields from non-linear
galaxy density fields to de-correlate the covariance in redshift-space galaxy
bispectra. In particular, we evaluate the covariance of the bispectrum for
massive-galaxy-sized dark matter halos with reconstruction by using 4000
independent $N$-body simulations. Our results show that the bispectrum
covariance for the post-reconstructed field approaches the Gaussian prediction
at scale of $k<0.2\, h\, {\rm Mpc}^{-1}$. We also verify the leading-order
PNG-induced bispectrum is not affected by details of the reconstruction with
perturbative theory. We then demonstrate the constraining power of the
post-reconstructed bispectrum for PNG at redshift of $\sim0.5$. Further, we
perform a Fisher analysis to make a forecast of PNG constraints by galaxy
bispectra including anisotropic signals. Assuming a massive galaxy sample in
the SDSS Baryon Oscillation Spectroscopic Survey, we find that the
post-reconstructed bispectrum can constrain the local-, equilateral- and
orthogonal-types of PNG with $\Delta f_{\rm NL} \sim$13, 90 and 42,
respectively, improving the constraints with the pre-reconstructed bispectrum
by a factor of $1.3-3.2$. In conclusion, the reconstruction plays an essential
role in constraining various types of PNG signatures with a level of $\Delta
f_{\rm NL}<1$ from the galaxy bispectrum based on upcoming galaxy surveys.
