If my Yveltal with "Fright Night" Ability gets KO'd, am I then able to use Exp.Share on one of my Benched Pokemon to recover one of Yveltal's Basic Energy cards?
Sorry, but no. Fright Night would still be working so Exp.Share would not enable before Yveltal hits the Discard Pile.
Source: TPCi Rules Team (2017-01-05)

Relevant cards

Note: All applicable cards may not be listed below.
XY5_128
XY8_94