If you were to open Console and examine system.log while the unresponsiveness occurs, you would almost certainly find log entries identifying an inability to read or write from the drive, but if the system becomes unresponsive you won't be able to copy those entries for further examination until after you restart the Mac. Once restarted, finding the relevant entries would need to be accomplished by examining the time stamps of those entries.
Having said that, if you were to run OS X on an external volume, and abruptly disconnect that drive while using the Mac, it would act exactly like you describe. If that were the case, examining system.log would be pointless because no log entries can be written to a drive that is no longer available (obviously). Perhaps that's the reason Apple could find nothing wrong.
In your case I believe there will be no progress toward a solution until you replace something in an effort to isolate the cause. The fusion drive could be defective, or there could be some other hardware fault involving a connection to it and the logic board. Determining the defective component will be a process of elimination.
The procedure for examining system.log appears below, but as I wrote the problems may occur after the losing the ability to write potentially diagnostic information to the drive.
Next time the system becomes unresponsive note the time in your Mac's menu bar or some other accurate timepiece. Then, force it to shut down as you have been doing.
Then, start the Mac the way you normally start it.
Then, find the Console app - it is in your Mac's Utilities folder and looks like this

You can find it by selecting Utilities from the Finder's Go menu.
Open Console.
If the log list column on the left is not already displayed, show the log list by selecting Show Log List from Console's View menu.
Locate system.log in the list and select it. Many date and time-stamped entries will appear, hundreds of them, and you must find the entries relevant to your Mac's problem.
To do that type the words bootlog in the Filter field at the upper right of the Console window:

That will cause all log entries to be hidden, except for the entries containing those words. The last entry in the list will correspond to the last time you started the Mac. It will have a time stamp in the form hh:mm:ss later than the time you noted in the Mac's menu bar prior to shutdown.
Make note of that exact time in hours, minutes, and seconds.
Next: Clear the filter field by clicking the x in the above screenshot. It should then look like this:

All previously displayed log entries will reappear.
Next: Find the log entry that corresponds exactly to the time you noted in hours, minutes, and seconds containing the words "BOOT_TIME". You may have to scroll up or down a considerable distance. When you find that line, select it and drag your cursor up and prior to the entry corresponding to the approximate time you noted before shutting down the Mac. This will result in a selection of perhaps 100 lines or so.
One or more of them, along with their time stamps, may reveal the reason for the unresponsiveness.
Copy and paste those log entries in a reply. Please don't post thousands and thousands of lines.
Most of the entries will be cryptic but will contain information you might consider personal such as your Mac's name. If you do not want that information to appear, delete or obscure it when posting your reply. Leave enough information so that the entries can be deciphered.