As in, if the Soviets got to control the ultimate "higher ground", they could rule the world from it. Apollo made sure no one could control space, and we now have peace in space.
I kinda believe in this, though I don't know how historically verified it is. It's of course not what you ever say in your rousing speeches to the nation and the world!
On that basis the military was advocating for a moon mission years before Kennedy's speech. It's hard to tell if this actually played a big role in the decision to go to the moon, but it was the cold war after all.
Also the Soviet Union was winning most of the relevant milestones in space (first satellite, first animal in orbit, first human, first woman, first space walk, etc), and this was a great source of national pride. Beating them had great propaganda value.
The Cold War resulted in a lot of really nutty thinking so I can't rule it out entirely, but it's hard to imagine what such a base would look like, especially with 1960s tech.
Of course actually getting ICBMs on the moon is a whole other endeavor.
You'd never get first strike launching from the Moon, regular ICBMs would be all over you long before the moon IPBMs made the 1 day journey back to Earth. Even with the faster travel speed from only having to accelerate out of the Moon's gravity well, you're talking about a distance of 384,000km vs. 8,000km. Your missiles arrive to a planet that is already a radioactive moonscape.
It would also be fairly obvious that someone was trying to move several thousand pounds of hardware onto the moon, and launch detection satellites would be even more effective at monitoring the moon. So you've got a first strike, that will be detectable basically immediately, and weapons that now need to travel ~385,000 km and survive even more serious re-entry than the ICBM's based on Earth. Or you've got a second strike capability that cost you billions of rubles/dollars.
Once you look at it like that, its pretty easy to see why the submarine launched weapons were the preferred method for threatening rapid first strike while providing a guaranteed second strike capability.
The lunar orbit->earth delta-v is 4.8km/s. I didn't check your other numbers but they seem to be off by a factor 1000 too.
I didn't do the math but at that speed you don't need a warhead to destroy a target. Let's try with 1000 kg steel or whatever won't melt down
1/2 mv^2 = 500 * (3200 * 3600)^2 = 345,500,000,000 J
1 J lifts about 100 g mass by 1 m on the surface of Earth. That energy would lift 345,500,000 kg by 100 m.
I would rather say that the Soviet Union was winning the irrelevant milestones, precisely for the propaganda value. First woman in space means nothing, really - you put a woman on your spaceship, wham, easy win. The moon landings forced the USA to tackle real engineering challenges - rendezvous and docking, piloted spacecraft and piloted orbit changes, functionally useful EVAs, keeping humans alive in space for long periods, etc.
One of the last major "firsts" was "first organisms to circle the moon" - they barely squeaked this one in a few months before Apollo 8, and they accomplished it by shoving some tortoises into a Soyuz (no food or water required!).
First woman in space is technically easy, but it does say something about the two cultures that the Soviet Union put a woman in space almost immediately, whereas the United States didn't do so for decades.