You will automatically go to the hideout after school, where you will decide on sending the calling card out or not. As Morgana points out, once you steal the Treasure, the Palace will go away, so you need to make sure you've done everything you wanted to before doing this. Also, sending out the calling card will have you return home for the day, then begin the next one at the Palace. Choose the option to send the calling card and then it will be evening once again, with you being at home (save here).
You can't do much of anything, but if you head up to your room, you will be able to make some infiltration tools. Hopefully, you have the materials needed to make two Lockpicks, as Morgana will allow you to make two of them. However, note that you could be lucky and get three total, but that might require you to reload your save. If you feel like doing this, then go ahead, but it'll help if you wish to get the remaining locked chests in Kamoshida's Palace.
At school the following morning, you overhear some students talk about your one shop at school having leftover yakisoba pan on Fridays. Unfortunately, if you're following this guide, then you won't have time for that. Watch the scenes that happen, then you'll automatically be at the Palace. Note that the Security Level is permanently set at 99% here, so if you wish to explore, you're going to have to deal with enemies being able to notice you easily and they instantly respawn.
When you are ready, fast travel to the Throne Room and make your way to the big double doors, which are now open. Climb the stairs in the back to reach the Treasure, where your Magician Confidant will rank up to level 2 and end up fighting the big boss of the dungeon.
BOSS - Suguru Asmodeus Kamoshida
For a first boss, Kamoshida isn't too bad, but he does have a few tricks up his sleeve that you have to watch out for. The fight is in two phases and during the first one, his most annoying move is probably Volleyball Assault, which can hit up to three times on a single character. Factor in that he gets two actions per turn and that damage can be troublesome, especially if he decides to target the same character twice. Not too far into the battle, he will use Libido Boost, which will heal him fully.
At this point, the trophy in front of him will become targetable and that will be your sole objective until it is gone. He will likely transfer to his other attacks during this phase, which will be a pain. Tongue Whip hits everyone in the party for around 30 damage at one hit, while Lustful Slurp will increase his attack, similar to Tarukaja. The latter will spell trouble, especially since you likely only have Dia to heal you. If you somehow got to level 11 during the dungeon, then Ann will have Tarunda, which can be used to lower his attack. If you see Kamoshida spend a turn looking at Ann, make sure you guard with her, as he will use Lick, a move that will likely do enough damage to kill her in one hit.
Once the Trophy of Obsession is destroyed, the second phase of the battle will begin after you do a scripted All-Out Attack. Kamoshida will gain a new attack called Golden Knife during this phase, which can hit someone for around 50 damage. This combined with Lustful Slurp can be big trouble, especially with the part coming up. You will receive a short tutorial about Special Orders, which will happen shortly after the battle resumes. You will need to decide to send someone after Kamoshida's crown, with that party member acting on their own and staying out of the battle for a while.
The best choice for this might be Ryuji, as both Morgana and Ann are useful to keep around for healing. Continue attacking him and when he announces his "killshot," have everyone guard, as he will execute his biggest move the next turn. Even when guarding, Gold Medal Spike does around 40 damage to the party. Keep this up for a few more turns and the person you sent will knock Kamoshida's crown off his head. This does two things, the first of which is reducing the attacks he gets down to one. The other thing is that he will take a ton more damage from your attacks, essentially making the battle a cinch.
Overall, Kamoshida's biggest worries will be the fact that he can attack twice per turn. He has no strengths or weaknesses, so use whatever you want. You will likely need to resort to using Dia with Morgana and Ann on each turn, unless he misses an attack. If you managed to fuse Mokoi, then Skull Crusher is a great attack to use on Kamoshida, especially if you buff up with Tarukaja (Ryuji has it). Stick with Lunge with Ryuji, unless you need to use an item to heal, and in the turns when the other two are free, you can use their elemental skills.
Alright! Things are finally over, as the Palace is now destroyed and you return to the real world. There isn't much you can do at Leblanc, so head straight to bed to see if anything has changed at school.