Is it possible for a moon to have the same orbital period as its planet? Is it possible for a planet to take just as long to orbit its star as a moon takes to orbit the planet? If we assume circular orbits, then
$\text{orbital period}\sim \sqrt{\frac{\text{radius}^{3}}{\text{parent mass}}}$
and thus what we need is
$\frac{r_{moon}^3}{m_{planet}} = \frac{r_{planet}^3}{m_{star}}$, or in other terms, $\left( \frac{r_{moon}}{r_{planet}} \right)^3 \simeq \frac{m_{planet}}{m_{star}}$
As far as I know, the only limiting factor on a moon's distance from its planet is that eventually the gravitational field strength of the star approaches that of the planet, pulling it out of orbit. This means that we need
$\frac{m_{star}}{r_{planet}^{2}} \ll \frac{m_{planet}}{r_{moon}^{2}}$, or in other terms, $\left( \frac{r_{moon}}{r_{planet}} \right)^2 \ll \frac{m_{planet}}{m_{star}}$
Unfortunately, doing the algebra on this leads us to conclude that $\frac{r_{moon}}{r_{planet}} > 1$ which cannot happen.
Is this the case, or have I overlooked something?
 A: A Klemperer rosette is sort of way that it could happen. Like @rob's Lagrange point idea, it is unstable.
Pick a rosette with many bodies of three types: Sun, Planet, Moon. Arrange them like this. S, P, M, P, S, P, M, P, ...
Because there are many bodies, the nearest are nearly in a line. Each Moon is hidden behind two Planets from two Suns.
A: This occurs at the Lagrange points, where the moon is stationary in the rotating reference frame. A moon at L1, between the planet and the star, or at L2, on the "outside" of the planet's orbit, has a sidereal period equal to the planet's year.
Unfortunately for your idea, L1 and L2 are unstable equilibria. The stable equilibria at L4 and L5 are much more like the moon and planet are co-orbiting the star than like the moon is orbiting the planet.
You might be imagining the case where the "synodic period" of the moon is equal to the planet's year.  For example, imagine that the full moon only happened in June, and the new moon only happened in December.  This corresponds to a resonant orbit where the sidereal period of the moon is twice the sidereal period of the planet. You can almost certainly come up with mass ratios where this resonance is allowed, and where the moon's orbit also fits inside the planet's "Hill sphere" where orbits are stable.
Your analysis of gravitational force balance is incorrect because it neglects the centrifugal force in the rotating frame.  In the limit of a low-mass planet, the radius of the Hill sphere is approximately
$r_\text{moon}^\text{max} = r_\text{planet}\sqrt[3]{\frac{m_\text{planet}}{3\cdot m_\text{star}}}$.
