Not sure if this has been mentioned, but one source of chirping is USB devices, wireless mice, etc., especially if they have a wireless receiver. Some receivers are sensitive to cell phones and will "chirp" when they send signals to check email, etc. Think about anything wireless in the vicinity (typically a few feet) of a receiver, as well as shutting off BT on the computer when in DAW mode.

I think mettelus' comments might be on the right track. 
I used to work as a technician in a distant education class, like a video conference between classrooms, (with remote control cameras in the classroom, along with desk mics on all the student desks.  Whenever a student's cell phone would go off, even if it was on silent/vibrate, I would hear a similar chirping in the control room speakers.  I'd have to interrupt the class and go around and make sure everyone's cellphone was not just on silent, but completely powered off. 

So I would check things like that.  Any wireless components, including USB wireless devices, cellphone, bluetooth devices.  Even if it's not close to anything, even if it's in the same room, it could create that chirping noise.

Just another thing to check.
 

Just an update, for those interested.
 
I wasn't able to reproduce the problem until today.  I had been recording vocals for about an hour without a problem. This was using the same signal chain listed earlier in the thread.   I then began recording a bass guitar direct into the MOTU interface.  It worked without a problem for 20 minutes or so.  I was recording, replaying, recording, etc.  Then the artifact started again.
 

• I recorded again, and it still occurred.
• Restarted SONAR, recorded again, still there.
• Restarted the computer, started SONAR, recorded again, still there.
• Finally, I restarted the MOTU and the problem went away.

 
Interestingly I would have expected it to be a driver / computer problem.  I'm surprised it is the MOTU.
 
It certainly wasn't the cables, the patch panels, or even the low-end gear (well, other than the MOTU).
 
Now I have to decide if I replace my interface.  I really like its browser-based mixing capabilities and ability to name the I/O ports.

Does not the Motu AVB use a wireless connection to talk to other devices like iPAds?  If so possibly disabling it? 

No, no wireless.  It did occur to me it could be a power surge of some kind, but it has been plugged into a conditioning UPS this whole time.  I'm reaching out to MOTU to see if they can help.  

Audio interfaces do fail. Frequently. I had a MOTU for years until one day it abruptly had no audio output all. They are all consumer-grade electronic devices, with many cost vs. quality engineering compromises. But that's the reason we can afford them in the first place.

I heard back from MOTU on Monday (about a week after posting in their tech forum).  I produced a much better example.  Their answer: the X99 chipset on my motherboard.  I figured I had ruled out my computer with the steps I took, but perhaps not.
 
To test that out, I ran a test on a difference computer (a laptop), and after 8 hours, it was working perfectly.  Last night I repurposed another computer I happened to have that is better suited for a DAW and so far it seems like its OK.
 
The good news is I don't have to spend any money.  The bad news is that I already bought the new (and better) MOTU 828es, but will probably send it back!