Send us resources or we will kill all your satellites, will make it impossible to launch new and will occasionally make it rain fire.
There are to main problems with this approach.
You cutting my satellites, Luna? Well, ❤❤❤❤ you too!
Earth can spray dust on Moon's with practically as much ease as the Moon can do it to Earth.
Remember, the Moon should need satellites and telescopes even more than Earth. Unlike the Earth, they have no atmosphere, so they must detect naturally incoming asteroids in time to react to them. This goes worst for the Moon than for the Earth.
The damage goes both ways
Hitting on Earth's satellites is a MAD tactic, and you'd have to be really mad to do it. See, the M in MAD stands for "mutual" in a very literal sense.
If the Earth loses its satellites, its communication infrastructure will have taken a major blow. And the Moon is as dependent on that as the Earth. If you lose all comms with Earth, you cause, for all practical purposes, an interplanetary version of a netsplit.
All cryptocurrency trade will have to halt, on risk of ❤❤❤❤❤❤❤ up all transactions done while the connection is split. Anything that gets routed through the Earth - such as connections to and from other planets will be lost. Any assets in Earth banks will be inacessible until the infrastructure is up again.
But most importantly, Twitter, Facebook, Instagram and all high-gravity (from lunarian's point of view) porn that is not backed up locally in the moon will be inacessible. This might cause uprisings.