While we're waiting for a proper solution for embedding Bible passages, I created a hacky userscript (my first ever). I've used it for almost a week now and am happy with it. Works on Firefox and Chrome. No warranty, etc.
The script inserts a textarea below the passage, looking something like this (screenshot taken before some bugfixes):
The textarea content is ready for copy-paste, and should look quite nice. Example:
Psalm 23 (MSG)
1-3 God, my shepherd! I don't need a thing.
You have bedded me down in lush meadows,
you find me quiet pools to drink from.
True to your word,
you let me catch my breath
and send me in the right direction.
4 Even when the way goes through
I'm not afraid
when you walk at my side.
Your trusty shepherd's crook
makes me feel secure.
5 You serve me a six-course dinner
right in front of my enemies.
You revive my drooping head;
my cup brims with blessing.
6 Your beauty and love chase after me
every day of my life.
I'm back home in the house of God
for the rest of my life.
Now, there probably are bugs. I won't promise to fix them, but you can try reporting here or in the Github issue tracker. Even better if you can provide fixes.