If I Knock Out my opponent's Snorlax Doll that has a Luxurious Cape tool attached, do I draw a Prize Card for it?
No. Snorlax Doll says that you cannot draw ANY Prize cards when it is Knocked Out.
Source: TPCi Rules Team (2023-11-09)

Relevant cards

Note: All applicable cards may not be listed below.
SV4_175
SV4_175
SV4_166
SV4_166