Can the First Movie promo Pikachu use Recharge to get a Rainbow Energy from the deck, or does it have to be a basic Lightning Energy card?
That would not work. Rainbow Energy only counts as all-basic energy types WHILE IN PLAY. This is not the case when it is in the deck.
Source: WotC Chat (2000-04-27)

Relevant cards

Note: All applicable cards may not be listed below.
HGSS1_104
WOTCPR_4
PL1_121