Topological superconductors have attracted wide-spreading interests for the bright application perspectives to quantum computing. Cu$_{0.3}$Bi$_2$Se$_3$ is a rare bulk topological superconductor with an odd-parity wave function, but the details of the vector order parameter $textbf{{d}}$ and its pinning mechanism are still unclear. We have succeeded in growing Cu$_x$Bi$_2$Se$_3$ single crystals with unprecedented high doping levels. For samples with $x$ = 0.28, 0.36 and 0.37 with similar carrier density as evidenced by Knight shift, the in-plane upper critical field $H_{rm c2}$ shows a two-fold symmetry. However, the angle at which the $H_{rm c2}$ becomes minimal is different by 90$^circ$ among them, which indicates that the $textbf{{d}}$-vector direction is different for each crystal likely due to a different local environment. The carrier density for $x$ = 0.46 and 0.54 increases substantially compared to $xleq$ 0.37. Surprisingly, the in-plane $H_{rm c2}$ anisotropy disappears, indicating that the gap symmetry undergoes a transition from nematic to isotropic (possibly chiral) as carrier increases.
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