I'm having the same problem with the Moogle shop while trying to exit from the Synthesize option. I'm using an M3 Real and I manually hex patched the ROM with the number 37 and got the game to work. I'm sure if updating my software firmware will help, but I've tried both the iTouch v.3.2e and Touchpod Firmware v. 4.4a X52 software and the game still doesn't work. Is there another hex code that we can use to manually fix the synthesize exit freeze just like that one hex edit to get past the difficulty selection screen?
Edit: Yes, I've been a member for 2 years and 5 months (minus 1 day) and have NEVER posted until today. So obviously, I'm kinda like Xion who never said anything until Day 24 of the game. ^^
Edit 2 (3:44 PM): Ok, I've actually found the solution to the problem myself actually. For anyone who doesn't use AKAIO 1.5, this fix is for you. I've only specifically got it to work on my M3 Real, but R4 users and anyone else should give this patch and process a shot.
Step 1: Download file from this page -
http://gbatemp.net/index.php?download=7261
Step 2: Download Lunar IPS program here -
http://fusoya.eludevisibility.org/lips/download/lips101.zip
Step 3: Place a clean Kingdom Hearts 358/2 Days into the folder that contains the lips101 program. You don't have to, but it makes it easier to find the ROM you're patching.
Step 4: put the ips from the "[7261]KH358_2D_U_.rar" file in the lips101 folder as well.
Step 5: Click the "Apply the IPS Patch" button and a window will appear.
Step 6: Select the "Kingdom Hearts 358-2 Days (U).ips" file click the "Open" button.
Step 7: Go to the box that says "Files of type:" and change the option from "Most Common ROM files" to "All Files (*.*)"
Step 8: Select your clean "4215 - Kingdom Hearts 358-2 Days (U)(Xenophobia).nds" ROM file and click the "Open" button.
Step 9: Move you're newly patched ROM to your Flash Cart and play the game WITHOUT the Moogle, Synthesize, exit, freeze bug. Enjoy!!^^
My sincerest thanks to Gokuguy for creating the one patch that fixes the new game and synthesize exit bugs.