This is a difficult issue. Let's see if there's something I can contribute.
The original and unlit 11' VFX model for "The Cage" just had a somewhat transparent lower dome
After the upgrades on the regular series model the dome
was lit and had a nipple or something added to it. The best VFX close-up footage
became available in "That Which Survives" and "Let That Be..."
I'll have to join the chorus of the skeptics. While we can clearly see that there is an opening or a dark dot on the nipple (the phaser bank/s in "The Doomsday Machine"?!), there is no visual indication for this tallywhacker. Don't let me be misunderstood: I won't categorically deny it might have been there, but the aforementioned Season Three close-up footage could indicate it had fallen off by the time this footage was shot.
Unfortunately, the close-up shots of the 11-footer in the Smithsonian are inconclusive as apparently the lower sensor dome had gone missing, too, and was replaced. At least the nipple would have to be there, but it's missing, so it's just another replacement part.