No it doesn't
The affects of Transform are;
Pokémon Species
Pokémon Cry
Attack Stat
Defense Stat
Speed Stat
Special Attack Stat
Special Defense Stat
Capture Rate
Stat Changes
EXP Yield
Pokémon Moves
Additionally, I just put in a Snorlax with full Happiness, no evs against a Ditto with Impostor, no EVs and 1 happiness.
Snorlax did 84% damage with return
Ditto did 1%
So yeah, transform does not copy happiness on a Pokemon.