I know of one mapping of a lemniscate of Bernouilli to a circle in the complex plane. The lemniscate $$ r^2 = 2 \cos(2\phi)$$ maps to a circle under the mapping $$z \mapsto z^2.$$
The lemniscate has its midpoint in the origin and foci $\pm 1$. The circle passes through the origin and has its centre at $+1$.
This is a mapping where all Cassini curves transform into concentric circles.
I found it on page 61, 62 of Tr. Needham's Visual Complex Analysis.
My question is: do you know any other mappings/functions that transform lemniscates or Cassini curves into circles?
Since you already have the $f:z \mapsto z^2$, any composition $g\circ f$ where $$g:\text{circle-to-circle}$$ gives you another mapping. In other words, given a mapping (any mapping from lemniscate to circle) that doesn't "look like" a composition using $z^2$, it is still actually equivalent to such a composition $g(f(z))$ with some $g$.