Actually, once you hacked the new system, you can preform the system transfer as normal
What you need to do first is make a NAND backup of your old system or dump the tickets using Decrypt9.
From there, do the system transfer. You will transfer all of your stuff to your new system, including your CIAs.
Restore the NAND back to the old system
Use FBI to inject the tickets from your old system to the new system. This will re-activate the CIAs that appear to be missing.
From there, use your old system as reference to the games that weren't able to be re-installed.