Your preference is for initial observation by Mr. Average Joe. We can see how plausible this is by creating some categories of how likely Average Joe is to have access to certain vantage points.
From most plausible to least plausible (list is not exhaustive):
- Surface of Earth - very easy, just walk outside
- Low Earth Orbit (hudreds of km up) - not as easy but still plausible; Average Joe can get viewing time on such telescopes
- Geosynchronous Orbit (10s of thousands of km up) - even less likely, possibly implausible; can Average Joe get viewing time on such satellites? Not that I know of, but I'm not an expert
- Lunar Orbit - Not unless Average Joe has some special job, connections, or story plot hook
- Something else farther away, such as Earth-Sun L-points instead of Earth-Moon L-points, or other non-Earth/Moon-bound objects - Not feasible for Average Joe
Now let's do some simple, rough trig estimations to see how far from Earth you'd need to be to see around the moon to that point.
Distances in thousands of km:
+ < Moon edge
^Moon center ^L2
All we have to do is scale that triangle up by multiplying the sides. We know that the distance Earth-to-Moon is ~384000km, so the distance Earth-to-Lunar-L2 is ~(384000+61000)km = ~445000km. That is a factor of ~7.3 increase from 61000km, so...
(again distances in 1000s of km)
+ < X (which is where we want to be at or past to see Earth-Moon-L2)
+ + < Moon edge ~2
^ Earth center ~445 ^Moon center ~61 ^L2
This has now become a "Solve for X" math problem.
X / 2 = 445 / 61 After solving for X by shifting the "/2" over, we have
X = 14
So that is 14000km from the center of the Earth, so for orbit distance about half that number of km from the surface of the Earth. A calculation that I did shortly before writing this answer put it at a 6000km orbit, and I think that quick calculation was slightly more accurate than this one.
Roughly, 6000km is about 10 times farther than low earth orbit, but is only about one-tenth as much as geosynchronous orbit. So Average Joe cannot view L2 from his backyard, nor from a low earth orbit telescope. Average Joe would need to beg, borrow, steal, or pay for time on something farther out. Possible, but not quite as feasible.
It might still work for your story if this is Academic Astronomer Joe or Wealthy Hobby Astronomer Joe rather than just Average Joe. Either that, or maybe a plot hook that gets Average Joe some experience and some time on the appropriate telescope. Plot hooks could include being a college astronomy major or having a friend or relative that can get Joe the telescope time.
As Joe rode his bike past the observatory, he saw the last car leave
their parking lot. Joe had wanted to get back into there ever since he
got a tour of the place for a 9th grade STEM event. They even had a
computer in there that has access to a telescope in geostationary
orbit that they got to play with for a few minutes - it was as if they
were sitting out in space and looking out at the cosmos!
Joe couldn't shake the feeling. Despite being on detention again
this week for breaking into the high school's science lab after hours,
he just could not pass this opportunity up. With the holiday tomorrow,
nobody would be back at the observatory for days. It's decided! The
only question now is where to hide the bike from street view for the
(break away to another set of characters, then next chapter come back
Joe could hardly believe it, the password was written down on a sticky
note in a drawer nearby! Didn't these guys ever get a computer
security lesson when they were in school? Oh that's right, that wasn't
a thing when they were Joe's age.
After punching in the password, Joe was in. Luckily, Joe was paying
close attention on the tour when the astronomers were controlling the
orbiting satellite. Hmm, it's trained on one of Saturn's moons - so
far away, so tiny, so pixelated... so boring. Let's look at something
closer. The moon should be easy to find. What is that tiny patch
there? It's black, but it still stands out, just barely...