快把大衣穿穿！When is it wrong?
Ludi wrote:


  according to our book, reduplication of the verb makes the predicate complex, fulfilling the requirement for (把)字句


Reduplication of verb imply the verb is in 'a slight degree', 'a casual manner' or 'tentative manner'

Examples:

吃 = eat

吃吃 =  (slightly/ casually) eat

试 = try

试试 = (slightly/ tentatively) try

The first part of the sentence '外邊很冷' is a reason; the second part of the sentence '快把大衣穿穿 is a request/ command, you can switch the positions of the 'request/ command' and the 'reason' and the meaning would be the same. '快把大衣穿穿, (because) 外邊很冷'

Since this sentence request/ command  the object to do something definite, it shouldn't be in reduplication form that implies it is in 'slight', 'casual' or 'tentatively' manner

user3a is correct:


  外邊很冷，快把大衣「穿上」


把 is not an important factor in choosing reduplicate the verb or not.
You shouldn't  reduplicate 穿 in "外邊很冷，快穿上大衣" neither.