The equation $|z-\omega|^2+|z-\omega^2|^2=\lambda$ will represent a circle if

We have that

$$x^2+y^2+x+1-\frac \lambda 2=0\iff\left(x+\frac12\right)^2+y^2=\frac \lambda 2-\frac 34$$

with the condition $\frac \lambda 2-\frac 34\ge 0 \iff \lambda \ge \frac 32$.

Just another similar way, from your equation $2(|z|^2+1)+(z+\bar z)=\lambda $, it is also possible to write directly that $$\tfrac12\lambda = |z|^2 + \tfrac12z + \tfrac12\bar z+1 = (z +\tfrac12 )(\bar z +\tfrac12 )+\tfrac34$$ $$\iff \left|z +\tfrac12\right|^2 = \frac{2\lambda-3 }4$$

Now that's the equation of a (non-degenerate) circle entered at $-\frac12$, iff RHS is positive, i.e. $\lambda > \frac32$.