In this answer, I’ll attempt to address two primary concerns impacting the habitability of your moon:
* atmosphere retention
* absorption of solar radiation
You will undoubtedly have to tweak the parameters of your planet in order to get the desired weather patterns. However, these two factors seem most important with respect to habitability.
Before getting into the weeds, here’s a list of definitions for variables I’ll use:
* $R_m$, the moon’s radius
* $M_m$, the moon’s mass
* $L_{s}$, the combined average luminosity of the three-sun system
* $D$, the distance of the planet-moon system from the three-sun system
* $G\approx 6.7\cdot 10^{-11} \space\text{Nm}^2/\text{kg}^2$, the gravitational constant
* $k\approx 1.4\cdot 10^{-23}\space \text{J}/\text{K}$, the <PERSON> constant
Alright, let’s go! (Note: I’m bound to have made some computational error somewhere below. Hopefully it doesn’t affect my estimates too much, and they’re still within the right order of magnitude. Bonus points if you find a mistake!)
Atmosphere retention
No matter how massive or cold your planet is, it will always continuously lose some of its atmosphere (as long as this atmosphere is gaseous). This is because not all of the atmospheric gas molecules have the same speed - their speeds are random, following the Maxwell-Boltzmann Distribution. At all times, some of the molecules will be moving fast enough to escape. 24
The question is - how long do you want your atmosphere to last?
The escape velocity for your moon is approximately equal to $$v_{\text{esc}} = \sqrt{\frac{2GM_m}{R_m}}$$ and the root-mean-square velocity of gas molecules in a gas of temperature $T$ is equal to $$v_{\text{rms}} = \sqrt{\frac{3kT}{2m}}$$ where $m$ is the mass of the gas molecule in question. You certainly don’t want $v_{\text{rms}}>v_{\text{esc}}$, or your whole atmosphere will be gone in an instant. So, at the very least, you need
$$\sqrt{\frac{3kT}{2m}} \lt \sqrt{\frac{2GM_m}{R_m}}$$
or, for a molecule of diatomic oxygen,
$$\frac{M_m}{R_m T} \approx 2.92\cdot 10^{12}\frac{\text{kg}}{\text{m}\cdot\text{K}}$$
For a moon the size of Deimos (which is almost certainly much smaller than yours) and with an average surface temperature equal to Earth’s, the LHS of this inequality is approximately $8.3\cdot 10^{8}$. That’s well below this rudimentary upper limit - so far, so good.
Let’s get a little more nitpicky. Remember what I said before about how some of your planet’s atmosphere will always be escaping?
Assuming the atmosphere’s depth is negligibly small compared to the planet’s radius, we have that the surface area of atmosphere exposed to space is approximately $4\pi R_m^2$. According to the Maxwell-Boltzmann distribution, if $T$ is the average temperature, then the proportion that have achieved escape velocity at any given time is equal to
$$\begin{align}\alpha_{\text{esc}} &= 2\sqrt{2\pi}\int_{\sqrt{GM_m m/kTR_m}}^\infty v^2 e^{-v^2}dv\ &= \frac{2\xi e^{-\xi^2}+\sqrt{\pi}\text{erfc}(\xi)}{4}\ &\sim \frac{\xi e^{-\xi^2}}{2} \end{align}$$ for reasonably small values of $\xi$, where $$\xi=\sqrt{\frac{GM_m m}{kT}}$$
As an estimate, let’s use the Moon’s mass and radius and Earth’s surface temperature (and consider diatomic oxygen molecules). This yields approximate values of $$\xi\approx 18.5$$ $$\alpha\approx 2. 24