I think Robittybob1 is sourcing
http://arxiv.org/abs/astro-ph/0407149
$$\begin{array}{ll} \textrm{Parameter} & \textrm{Value} \\ \hline \\ a_p \, \sin \theta_i & 2.3417725 \pm 0.0000008 \, c \cdot \textrm{s} \\ e & 0.6171338 \pm 0.0000004 \\ P_b & 0.322997448930 \pm 0.000000000004 \, \textrm{d} \\ \dot{P}_{b,Obs} & ( -2.4184 \pm 0.0009 ) \times 10^{-12} \\ \hline \\ m_p & 1.4414 \pm 0.0002 M_{\odot} \\ m_c & 1.3867 \pm 0.0002 M_{\odot} \\ \hline \\ c & 299792458 \textrm{m} \cdot \textrm{s}^{\tiny -1} \\ G & ( 6.6738 \pm 0.0008 ) \times 10^{-11} \textrm{m}^{\tiny 3} \cdot \textrm{kg}^{\tiny -1} \cdot \textrm{s}^{\tiny -2} \\ M_{\odot} & ( 1.9885 \pm 0.0002) \times 10^{30} \textrm{kg} \\ \frac{2 G M_{\odot}}{c^2} & 2953.2500770 \pm 0.0000002 \textrm{m} \\ \hline \\ \dot{P}_{b,Gal} & ( -0.0128 \pm 0.0050 ) \times 10^{-12} \end{array}$$
where the masses of the stars are derived from the primary observational values, and the last term relates to the geometry of the line-of-sight.
$$a_p$$ is the semimajor axis of the pulsar's orbit about the center of mass
$$\theta_i$$ is the angle of inclination relative to the line of sight
$$e$$ is the eccentricity of the orbit
$$P_b$$ is the period
$$\dot{P}_{b,Obs}$$ is the rate of change of the period
$$m_p$$ is the mass of the pulsar
$$m_c$$ is the mass of the companion star
$$\dot{P}_{b,Gal}$$ is the observational effect on the period due to the geometry of the light of sight
$$c$$ is the speed of light, which is exact in the SI units
$$G$$ is the gravitational constant
$$M_{\odot}$$ is the mass of the sun
Note that $$2 G M_{\odot} / c^2$$ is known to much better precision than either $$G$$ or $$M_{\odot}$$.
The contention of the paper is that
$$\dot{P}_{b,Obs} - \dot{P}_{b,Gal} = \dot{P}_{b,GR} = - \frac{ 4 G^2 m_p m_c }{c^4} \sqrt[3]{\frac{\pi^8}{P_b^5 c^2 (2 G m_p/c^2 + 2 G m_c/c^2)}} \frac{192 + 584 e^2 + 74 e^4}{ 5 c \sqrt{(1-e^2)^7}}
= -( 1.4414 \pm 0.0002 )(1.3867 \pm 0.0002)( 2953.2500770 \pm 0.0000002 \textrm{m} )^2
\quad \quad \times \sqrt[3]{\frac{\pi^8}{(0.322997448930 \pm 0.000000000004 \, \textrm{d} )^5 (299792458^2 \textrm{m}^2 \cdot \textrm{s}^{\tiny -2}) (2953.2500770 \pm 0.0000002 \textrm{m} ) (( 1.4414 \pm 0.0002 )+(1.3867 \pm 0.0002))}}
\quad \quad \times \frac{192 + 584 (0.6171338 \pm 0.0000004)^2 + 74 (0.6171338 \pm 0.0000004)^4}{ 5 (299792458 \textrm{m} \cdot \textrm{s}^{\tiny -1}) \sqrt{(1-(0.6171338 \pm 0.0000004)^2)^7}}
\quad \quad \times \sqrt[3]{\frac{ (1 \textrm{d})^5 }{ (86400 \textrm{s} )^5}}
= ( -2.4020 \pm 0.0004) \times 10^{-12}$$
That the authors use a smaller error quite possibly indicates they know that the errors of the estimated masses are anti-correlated in some way. But because the errors in observation are swamped by uncertainty in knowledge of the geometry of the line of sight, any attempt to figure out G from these observations will be affected by these errors which Robittybob1 seems to ignore as well as not communicate his procedures very well.
See also
http://pdg.lbl.gov/2012/reviews/rpp2012-rev-astrophysical-constants.pdf for other values.
See
http://adsabs.harvard.edu/full/1982ApJ...253..908T for the mass formulas which properly should be inserted into the above relation.