This trophy can be missed since the last palace takes place during the third semester.
First, you need to get
Dr. Maruki's (Councillor) Confidant to Rank 9 before he leaves on November 17th (when he's first introduced, Shujin staff mention that he'll only be there through November). On November 17th, his confidant will automatically progress to Rank 10. Like most confidants, the best way to rank him up is to have a Councillor Persona in your stock before hanging out with him as this gives you more points towards hitting the next confidant rank. Getting Maruki to rank 10 is important because this is what triggers the third semester, otherwise the game will end before the third semester begins.
Assuming you trigger the third semester, at four times during the story you will be offered different deals; the dates the deals are offered on are
You need to
reject all four deals.
Other than those two criteria, it's simply a matter of completing each Palace within the time limit.