Just an angle that hasn't been pointed out before:
In the prologue, it says
When urban legends were manifesting in Gensokyo, a rumour started to spread that "Apollo never went to the Moon".
Maybe this particular urban legend (moon landing hoax) also manifested, and the invasion is the result of this alternative timeline.
The Lunarians likely took offence when humans started exploring beyond Earth (Mars, in this case), and took Curiosity as retaliation. And because Apollo landing in their backyard never happened---the event of which shook them and which they treated with much paranoia---they probably thought less of Earth's capabilities than if they (Apollo) had. In Touhou's timeline, this would mean only the Chang'e program occurred, and this was what ultimately set them off.
Kaguya and Eirin's conversation implies that they knew the invasion was a possibility. I'm guessing that the invasion was proposed before, but due to fear/reluctance/hesitation, it didn't fall through until now.
So, my thoughts on what happened:
- Perpetrator wants to invade the Earth, but due to the Apollo program, higher authorities were against it.
- Perpetrator manipulates Sumireko with the Lunar Capital power stone, and takes advantage of the existing faked moon landing conspiracy theory.
- Apollo never
happened, and so the invasion was a go.
*Inserts remark about Moon stuffs being complicated, and curses insomnia.*