= Solution
Write $\phi=x+iy$. The added term is $-(x^2-y^2)$, so the full potential simplifies to
$$
V(x,y)=-2x^2+e^{\lambda(x^2+y^2)}.
$$
The phase-rotation subgroup is reduced from <U(1)> to the two-element <symmetry> $\phi\mapsto-\phi$. More fully, the kinetic term and potential are invariant under independent reflections $x\mapsto-x$ and $y\mapsto-y$, a $\mathbb Z_2\times\mathbb Z_2$ internal <symmetry> generated by $\phi\mapsto-\phi$ and <complex conjugation>. There is no continuous internal rotation <symmetry> for the <explicit twofold anisotropy of a complex scalar field>.
For fixed $x$, positive $\lambda$ makes the exponential grow with $y^2$, so every global minimum has $y=0$. On the real axis,
$$
\frac{\partial V}{\partial x}=2x[-2+\lambda e^{\lambda x^2}].
$$
Thus
$$
\boxed{\begin{cases}
\phi=\pm v,\quad v^2=\ln(2/\lambda)/\lambda,&0<\lambda<2,\\
\phi=0,&\lambda\geq2.
\end{cases}}
$$
For $0<\lambda<2$, the nonzero stationary points are the two global minima; the origin is unstable along the real direction. For $\lambda\geq2$ the origin is the unique minimum, with zero real-direction quadratic curvature at the critical value $2$. Selecting $+v$ breaks the sign reversal that exchanges the two vacua, while <complex conjugation> remains unbroken.
Set $\phi=v+\chi$, $\chi=u+iw$. Since $e^{\lambda v^2}=2/\lambda$, expansion gives
$$
\boxed{V=V_{\min}+4\lambda v^2u^2+2w^2+O(\chi^3)},\qquad
V_{\min}=\frac2\lambda\left[1-\ln\frac2\lambda\right].
$$
Equivalently, in canonical fields $h=\sqrt2u$, $\pi=\sqrt2w$,
$$
V=V_{\min}+\frac12m_h^2h^2+\frac12m_\pi^2\pi^2+\cdots,\qquad
\boxed{m_h^2=4\lambda v^2=4\ln(2/\lambda)>0,\quad m_\pi^2=2>0}.
$$
Both masses are positive throughout the broken <phase> $0<\lambda<2$. A broken discrete <symmetry> does not demand a massless <Goldstone boson>. At $\lambda=2$ the radial <mass> tends to zero at a <phase> boundary, so a claim of strictly positive <mass> there would be incorrect. For completeness, expanding around the symmetric origin gives squared masses $\lambda-2$ and $\lambda$ for the canonical real and imaginary fields.
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