The Moon Disk is a monster that hurts regional income (-20) but does not conduct raids. Fighting it can be a challenge due to its substantial HP (it can absorb several ships' loads of ammunition without dying) as well as its highly damaging attacks.
As with most monsters, having a special-teams ship like a flamethrower or a saw-blade can help, though since the Moon Disk tends to operate at 200 meters or so, your special-teams ship needs to have that sort of service ceiling. Unlike most monsters, though, the Moon Disk's means of attack reward custom-building a ship specifically for the purpose (and possibly scrapping it once the Disk is slain, to save on maintenance). Specifically, the Moon Disk shoots a slow-moving but extremely damaging (slightly more damage than the Imperial Cannon, roughly 10-meter blast radius: no point trying to armor against it) projectile, with a fairly slow rate of fire as compared to shipboard weapons. As such, the most cost-effective way to defeat a Disk involves having an Expensive ship that can move quickly (both speed and, if you're using command delay, quick command response). The Moon Disk is pretty good at identifying & aiming at the most expensive ship: this being your highly-mobile ship, maintaining command and keeping track of shots lets it dodge the Moon Disk's fire, allowing the rest of the fleet to pummel the Disk.
If you're using launched-from-mothership Aircraft (distinct from Airships), the Moon Disk generally doesn't target them; Dragonriders, being a small, fast, and expensive Airship for game purposes, tend to draw the Disk's attention. Though Dragonriders are certainly nimble enough to dodge the Disk's fire, they need to come rather close to effectively engage the Disk, which increases the risk that you'll slip up and miss a dodge, killing the Dragonrider. Bombers and Air Hussars can fly around and attack the Moon Disk without apparent danger, despite the in-game lore regarding the Moon Disk's noxious fumes. (Torpedo bombers are a bad idea, as they'll need to land and rearm. Conventional bombers will stay on station and continuously bomb the Disk.)
They're Big. They're Bad. But you can beat them!
Common: 2000 Research points (however, top-tier research caps out at ~1500)
Rare: Moon Disk Fragments bonus, enables use of Moon Disk Fragment Chambers