Let as assume there is a Schroder function with a fixed point for $f(z)$, with $|\lambda|>1$ for the fixed point. It doesn't matter if the fixed point is real or complex, but if $f(z)$ is entire with this restriction, then the inverse of the Schroder function is also entire. There is also a Schroder equation for $|\lambda|<1$, but its inverse is not entire since $f^{-1}$ is not entire, and solutions with $|\lambda|=1$ can be very complicated. The Schroder function, with fixed point $z_0$, is a formal series, $\psi(z)=z+a_2 z^2+a_3 z^3..$ defined by $\psi(f(z)-z_0) = \lambda \psi(z-z_0)$, and the inverse of the Schroder function has $\psi^{-1}(\lambda z)+z_0= f(\psi^{-1}(z)+z_0)$. Then one possible starting point for an analytic solution for a particular value of z, using the op's notation for g is:
$g(x,z)= \psi^{-1}(\psi(z-z_0)\lambda^x+z_0)$
$g(0,z)=z$
$g(1,z)=f(z)$
$g(x,z)=f(g(x-1,z))$
However, the op also requires that for some real valued c, $g(c,z)=\alpha$. So, the algorithm is calculate $y=\log_\lambda \frac{\psi(\alpha)}{\psi(z)}$. Now, y could be a real number, and in that case we are done since we can choose c=y, since $g(y,z)=\alpha$. But if y is not real valued, then we can easily arrange some analytic 1-cyclic function, $\theta(z)$, such that $\theta(0)=0$, and for any completely arbitrarily chosen real number c that is not an integer, $c+\theta(c)=y$. This would seem to meet the op's requirements.
Then one solution to the op's problem is:
$g(x,z)= \psi^{-1}(\psi(z-z_0)\lambda^{x+\theta(x)}+z_0)$
$g(0,z)=z$
$g(c,z)=\alpha$
Note, this solution works for z and $\alpha$ being real numbers, but $g(x,z)$ may not be real valued for other values of x other than 0 and c. If the op is interested in a function that is real valued for real valued x, then this solution is not sufficient. If $g(x,z)$ is real valued for all integers>=0, and $f(z)$ is a real valued function, and $g(\lfloor(c))<\alpha$ and $g(\lceil(c))>\alpha$, then a very creative solution involves generating $\theta(z)$ via a Riemann mapping of the interval between floor(c) and ceil(c), so that the function $g(x,z)$ becomes real valued, and nicely behaved at the real axis and in the complex plane. Actually, the Riemann mapping itself is messy, and usually involves a singularity or branch point. But basically, calculate a Riemann mapping of $\exp(2\pi i g^{-1}(z))$ for the real interval $[z..f(z)]$, and use this Riemann mapping to generate a $\theta(x)$, which can be used in the above equation, with one particular value of c, such that $g(c,z)=\alpha$. I think this is equivalent to perturbed Fatou Coordinates. I can try to give more details if this real valued function with a complex fixed point is what the op is interested in.