In Harry Potter Hogwarts Battle, can we use the Transfiguration proficiency when the Basilisk is out
Yes. In many cases like this the right answer can be found by looking for "keywords". In this case the keyword is "draw": the Basilisk does not allow you to "draw" additional cards. If an ability does not use the word "draw" then it is allowed even if it involves adding a card to your hand. In this case, "searching" your deck or discard pile and "adding" a card to your hand is not "drawing" an additional card as the term "draw" is not used. So the answer to the question should be Yes, you can use Transfiguration when the Basilisk is out.