If you assume, as I did, that we're talking about a United Earth, read from this point. If you carefully read the OP's post, which I finally did, skip to "EDIT" and read from there.
The historian Shelby Foote while speaking about the U.S. Civil War once concluded that the Southern Confederacy never had a chance. The Northern Union with its greater manpower and industrialization fought the war "with one hand behind their back." Had they suffered greater losses, they simply would have brought the other hand out.
Your lunar colony has the same problem in spades. No fundamental resources and dependent on Earth for almost everything (no matter how much handwaving is asserted, there's no such thing as 100% recycling and the next baby born would require new resources)... no matter how many weapons had been stockpiled, the moon would fight a valiant war and die in a blaze of glory... and would then be forgotten as a darkened cinder in the sky.
If it appeared the lunar colony was winning, the Earth would simply bring the other hand out... and then another hand... and another hand... having basically unlimited resources compared to the moon.
Further, it's unbelievable Earth would be so blind, so ignorant, that it would allow the moon to stockpile enough weapons to actually invoke Mutually Assured Destruction. Someone in the Earth bureaucracy would eventually say, "hmmmm.... you've been importing about 100X the putonium you need for your nuclear reactors... care to explain, or should we send in troops now?"
Now... if the Moon had somehow developed agriculture and asteroid mining... that might (might) change things, but it's still a bit hard to believe. Why wouldn't the greater manpower and governmental interference of Earth be watching all of that very closely?
EDIT: @JeffUK caused me to reread the OP's question and I must apologize that I misunderstood a critical point: I jumped to the conclusion the colony was defending itself against a united Earth — which for the reasons I state above I believe would be impossible. In reality, the Earth still has nations with competing interests and the colony will be attacked by a coalition.
(If, after reading the rest of my post you think a defense against a United Earth is possible, remember... unlimited resources. Missle after missle after missle until all the defensive tech is exhausted.)
MAD is irrelevant
In this situation Mutually Assured Destruction is irrelevant. If the parent nation cannot sustain it, the addition of the colony will not create it. The parent nation will always have greater and more expedient access to resources.
Besides, we're forgetting that missles take time to get to the moon. Using this somewhat unscientific assertion that missles travel at 4 miles-per-second (14,400 mph, I was too lazy to prove the article's veracity) they would require 16.6 hours just to get to the moon — or, more importantly, to get from the moon to Earth. That's a lot of warning.
It's easy to shoot down a missle if you have the time
Which brings me to my "new answer." Defending the moon from Earth-launched missles (or the Earth from moon-launched missles) is trivial (OK, it's not, but you'll see what I mean). Remember, the difficulty we have on Earth is that we have a window of only minutes to destroy the missle.
But lunar-bound or -launched missles have hours and hours. That's a long time to track trajectories, to launch counter-missles, and for those counter-missles to loop around a few times before they actually hit the incoming ETBM (extra-terrestrial ballistic missleTM :p).
As I think this new situation through, stealth would be critical, which means you need a means of propulsion that doesn't create a heat bloom that can be seen from Mars. The worry isn't a full-out war, it's a first-strike. A patient enemy may decide to declare war only after the two months they patienty took placing a hundred stealth missles in near-lunar orbit.
Now the moon has the same problem Earth has, minutes to fix the problem. If they can't stop the missles on Earth, they can't stop them on the moon.