iPhone Unable to be Unlocked

My iPhone 4S recently decided to make a dive out of the depths of my dresser when the need arose for my carrying a last-gen device to use for development testing. It's on iOS 9, but 9.0 only. And I can't unlock it.

I set a passcode before I put it away for its "nap", one which I stupidly forgot to notate anywhere, and one which is unique to the device. When I finally got it out and charged again, I had to go through the iterations of guessing many times until finally it locked me out of the device, telling me to "Connect to iTunes". I figured that once it established a handshake with a familiar device, it would unlock and at least allow me to restore. No such luck.

I've looked around, and the best advice I could find would be to put the device into recovery mode and restore it as a new device. I was able to do this, and it restored as I would've expected. It didn't, however, recognize the device. It restored, and even if I chose to set up as a new iPhone in iTunes, the screen was still locked.

I'm on the same Mac that I've used to sync since iOS 6, so nothing has changed hardware-wise. The computer almost doesn't recognize the iPhone, even though it used to be my daily driver and I used to sync nightly.

Is there any way to totally factory-wipe the iPhone to the point where I have to fully reload the software? I'm not for the life of me ever going to guess the password (not that I'm even given the chance anymore). I'm out of AppleCare or I'd take it to the Genius Bar.

Any help appreciated