I'm almost at the part where I face Shadow Lugia, and I know the natures are set the first time you encounter them before purification. I assume you have to save at the end as well, which frustrates me. So is there any way to look at the nature without saving? Yes, I know I have to battle with Shadow Lugia on my team against Greevil, but the nature is "???"

I'm thinking of losing on purpose when against Greevil to get Lugia's heart gauge low enough to reveal the nature. Or does this not work and I don't keep the shadow Lugia when losing?

well actually

You have to catch it
then at the purify chamber the tempo must ALL be max out in each of the boxes (I call them boxes)
If you did it right it should say
"lugia is responding!" or something like that

Well the nature is reveal when the heart gauge has 2 full white boxes. So what you're saying is that you're not really sure if I keep the Lugia after losing right?

Note: I saw my shadow Altaria's nature when it has 2 white boxes in the heart gauge.

If you were paying attention to the in-game text
the admins say XD0001 cannot be purified
so no mater what you do it heart gauge WILL NOT go down
the only way to find out its nature is to purify it in the purifying chamber

Then I just have to get lucky then. Thanks.

umm
no XD

you do NOT have to catch all the shadow pokemon
but its best to catch as many as you can, seeing how you have more options to bring up max tempo