Go Beyond!

If you click the quest item repeteadly and fast enough the quest will credit you for 2 or more captured moths. A favorable bug it seems.

I don't seem to be getting the moth net. I've abandoned the quest and reacquired making sure to have room in my bags!

edit: Ok so I traveled here to the beyond and couldn't find the net so I went back and got the quest again, not abandoning it until I was back near the questgiver.
2nd time I had the quest I came to the beyond and still didn't have the net so I was going to try to go back and try abandoning and reacquiring the quest to try again - this time when I walked over the mushroom travel area the quest progressed to having gone to the beyond and I now have the net so

TLDR: if you don't have the net try walking on the mushroom ring to see if the quest will complete step one in your progress and reward you the net

I used a macro to target moths and use the net.
I'd always spam the macro a few times. More than once, I would get 2x Selenia Moth in a spot where I could only see one.

When I did this at one spot in particular, without moving away, I was able to get six moths this way (repeating the macro).
/way 19.91 37.75
It might be possible that this works in any spot that you catch a moth, but it worked for me in that spot.

Even if that isn't a "thing" and / or stops being one, the macro makes life easier.

/target Selenia Moth
/run SetRaidTarget("target", 2)
/use Moth Net

The 2nd line isn't necessary for the macro to target and catch the moth. It just helped to indicate where the closest moth is, if you can't see it.

EDIT:

That thing, of standing in one spot, after you catch the first moth, of spamming the macro? It worked again.
I stood where I caught the first one and, after I had caught it with the macro, I kept spamming the macro and caught one after the other and did all of them.
It didn't seem like I was close enough to any of the other moths but, somehow, I kept getting credit for catching them.

Use that / a similar macro, put it in an action bar, spam it to catch the first moth and keep spamming it. You stand a good chance of catching all the moths in one spot / at one attempt.