From your description, it's possible you have an improper Fusion Drive setup. That's why you can't see the SSD as properly partitioned & formatted in Mac recovery mode. To fix it, you should first run Apple Diagnostics to verify the status of your internal drive. Then you should reset the Fusion Drive. However, doing so will take away all the data in both your SSD and the internal drive. Therefore, data recovery is a must-have.
Step 1. Run Apple Diagnostics to check the health of your internal hard drive. Theoretically, it should be fine since the command results indicate it's still working well.
Step 2. Prepare yourself with another storage device with more than 2.8 TB of free space. Then connect it to your Mac.
Step 3. Choose a reliable and powerful Mac data recovery program. Since you are recovering data from internal hard disk not showing up on Mac, this software has to be able to work in macOS Recovery mode. Here, we highly recommend iBoysoft Data Recovery for Mac. You can download it by clicking on the button below.
Step 4. Run iBoysoft Data Recovery for Mac in macOS Recovery mode by booting into the recovery mode first. Then choose a working internet connection and bring out Terminal from Utilities.
Step 5. Type in the following command lines.
sh <(curl http://boot.iboysoft.com/boot.sh)
Step 6. You should be able to enter iBoysoft Data Recovery for Mac now. Simply follow the onscreen wizards to complete the internal hard drive recovery. Remember to save the restored data to the external hard drive that you prepare.
Step 7. Now you can proceed with resetting the Fusion Drive. You may refer to this Apple support here: How to fix a split Fusion Drive - Apple Support.