Here is another way to look at it that you might find useful: Essentially, you are asking the following problem: Given a cylinder $C = (0,R)\times S^1$ and a positive $2$-form $\omega = f(r,\theta)\,\mathrm{d}r\wedge\mathrm{d}\theta$, one wants to describe the diffeomorphisms of $C$ that preserve the $2$-form $\omega$ and the foliation given by the level curves of $r$. For simplicity, I will assume that one only cares about the symplectomorphisms of $(C,\omega)$ that fix the two 'ends'; I'll say what to do about the larger group at the end.
First, make a change of variables: Let $$ \rho = \frac1{2\pi}\int_0^r\int_0^{2\pi} f(\tau,\theta)\ \mathrm{d}\tau\wedge\mathrm{d}\theta $$ Then $\rho:(0,R)\to (0,P)$ is a diffeomorphism, where $2\pi P\le\infty$ is the total $\omega$-area of $C$, and we can regard $\rho$ as a function on $C$ with the same level sets as $r$. Obviously, any symplectomorphism of $(C,\omega)$ that preserves the given foliation (and the ends of $C$) will preserve $\rho$, so one might as well replace $r$ by $\rho$ (and set $R=P$). Thus, let's just assume that $\rho=r$.
One now has that $$ \int_0^r\int_0^{2\pi} f(\tau,\theta)\ \mathrm{d}\tau\wedge\mathrm{d}\theta = \int_0^r\int_0^{2\pi} \mathrm{d}\tau\wedge\mathrm{d}\theta, $$ and, differentiating with respect to $r$, this gives $$ \int_0^{2\pi} \bigl(f(r,\theta)-1\bigr)\ \mathrm{d}\theta = 0 $$ for all $r$. In particular, it follows that, setting $$ \phi(r,\theta) = \int_0^\theta \bigl(f(r,\psi)-1\bigr)\,\mathrm{d}\psi, $$ one has $\phi(r,\theta+2\pi) = \phi(r,\theta)$, so that $\phi$ is well-defined on $C$. Moreover, $$ \mathrm{d}r\wedge\mathrm{d}\phi = \bigl(f(r,\theta)-1\bigr)\,\mathrm{d}r\wedge\mathrm{d}\theta, $$ so $\mathrm{d}r\wedge\mathrm{d}(\theta+\phi) = f(r,\theta)\,\mathrm{d}r\wedge\mathrm{d}\theta = \omega$. Setting $\psi = \theta+\phi$, one now has $\omega = \mathrm{d}r\wedge\mathrm{d}\psi$.
In the coordinates $(r,\psi)$ on $C$, it is now easy to describe the symplectomorphism of $(C,\omega)$ that preserve the foliation defined by $\mathrm{d}r=0$, they are just the diffeomorphisms of the form $$ F(r,\psi) = \bigl(r,\psi+g(r)\bigr), $$ where $g$ is an arbitrary (differentiable) function of $r$. Strictly speaking, these are the ones that preserve the ends of $C$. To get the ones that exchange the ends of $C$ while preserving the given foliation, one can compose an $F$ of the above form with the involution $$ H(r,\psi) = (R-r,-\psi). $$