So, mini guide for rooted android 13 for anyone facing issues like me:
Install Core Patch module for LSPosed and enable everything at it's UI.
Remove any magisk/LSPosed module you had previously installed with LuckyPatcher (if any). You can do this in LuckyPatcher.
Reboot your phone.
You'll now be able to install unsigned app but it might crash instantly when open if you're unlucky like me.
If that's the case. You'll need to sign the unsigned .apk like the quoted post mentions.
Then, make sure to open unmoded Blade Idle, login with your google play account and play for like 10 seconds or so.
Exit the game, force stop (close app) by using lucky patcher (or maybe even regular app info from android, not sure).
Install the signed modded apk. You'll be able to enter the game and your login info won't be lost until you turn off your phone, at which point you'll have to uninstall, reinstall the unmmodded, login, force stop and install the signed modded version again.