It is very, very hard to tell, but it hardly matters
Mesklin is so oblate that Newton's shell theorem stops applying: even according to Hal Clement himself that "To be perfectly frank, I don't know the exact value of the polar gravity; the planet is so oblate that the usual rule of spheres... would not even be a good approximation." Calculating the time dilation value for gravity sources near spheres is easy enough, but around Mesklin, not only is the shape not spherical, but the Lense-Thirring effect begins to kick in due to the planet's extremely high angular momentum. I work extensively in real life with the Lense-Thirring effect, and it gets complicated as the angular momentum of the central body increases.
Then again, what communications network? It's not like any human-built probe could possibly survive near the surface of the poles.
Okay but that's not fun
Okay, fine, I'll use a Schwarzschild approximation. Assuming that the mass of Mesklin is 16 Jupiter masses (about $3.0368\cdot10^{28}\text{ kg}$) and has a polar radius of $31,770\text{ km}$, then we can use the elements of the Schwarzschild metric to calculate a time dilation factor:
$$k=\sqrt{1-\frac{2GM}{rc^2}}\approx\sqrt{1-\frac{2\cdot6.6734\cdot10^{-11}\cdot3.0368\cdot10^{28}}{3.1770\cdot10^7\cdot8.9875517874\cdot10^{16}}}=0.999999290155.$$
For an observer distant from Mesklin, they would perceive a clock on Mesklin's pole ticking through 709 fewer nanoseconds every second than their own Mesklin-distant clock. That's about 61.3 milliseconds per day.
What does that mean for communication networks?
Oh, there are no communication networks in orbit. See, the Lense-Thirring precession I mentioned earlier changes the way orbits work around the central body; any communications satellites near Mesklin will observe a precession of their line of nodes (a critical-to-know astrodynamic value that essentially determines where on the planet the satellite passes over), and their time and space will also be screwed up. At first it might not be an issue, but over time one will observe that the satellite's orbit becomes erratic, and it might become impossible to keep it in a stable orbit.
Additionally, it's not like there's much down there to communicate with. 3 g of surface gravity makes it virtually impossible for modern-day chemical rockets to lift stuff off the planet and makes it hell to live there. If there absolutely needs to be communications networks going across the planet, they're definitely not going to be at the poles, so worrying about that shouldn't matter. Even if it did, it wouldn't mean that much: 61.3 milliseconds a day sounds bad, but for humans that kind of time dilation is so far below noticeable that nobody will even care, and for computers it just means changing a few values to synchronize the transmitters and receivers.
