When putting Banette into the Lost Zone with its "Puppet Offering" Ability, do you also put the underlying Shuppet card into the Lost Zone too?
Yes, the underlying card is part of the Pokemon; it is not "attached to" the Pokemon.
Source: Lost Origin FAQ; TPCi Rules Team (2022-09-08)

Relevant cards

Note: All applicable cards may not be listed below.
SWSH11_73
SWSH11_73