UPDATE:
Finally got the chance to take the GTX out today for a decent trial run.
The wife and I climbed aboard and headed directly to town to feed the thing - and don't you know the moment we got up around 50mph, I heard that same sickening, familiar sound that started this whole mess.

It's not as loud as before, but it's there again....
My wife starts listening around and fooling with things and announces that she doesn't think the noise is coming from the floor at all.
Instead, she says it sounds like it's coming from inside the dash somewhere.
By that time, we had reached the gas station, so I threw some gas in the car and reached under the hood to shut off the valve I have installed on the heater hose since it was a nice day and no heat was needed.
I climb back aboard and she's messing with the heater controls, asking why they all "stick".
I inform her the heater box is a mess and the blend doors are rotten, that the cables don't work very well, etc.
I then work each of the controls a bunch back and forth and shut off the fan.
We then set off again, as the plan was to take a drive up the highway to the next town (about 12 miles) to a store up there.
We get back up on the 4 lane and wouldn't you know - no noise.

The shifter needs some adjusting and the clutch pedal rattles going down the road (apparently Mr. Mechanic forgot a clip somewhere) but everything seems to be fine, so we enjoy the drive to the next town at about 60mph. It's in the low 70's and the car is running nice and cool.
We get to that store and park up front, which put the car in a nose-up incline position in the parking spot.
We're in the store about 20 minutes.
Back in the car, I go to start it up and the thing backfires through the carb!

(Keep in mind, the car has run beautifully since the engine swap and my fixing up the reman Edelbrock 750 on it.)
I crank it over again and it starts, but sounds like it's missing on a cylinder and the idle has decided to race up towards 1200rpm on its' own (it's usually 750-800rpm). This is totally different from how it ran on the way up there and totally different from how it has
ever ru.
I jump out, take off the air cleaner expecting to see the choke slammed shut or something - nada.
Check all the spark plug wires, etc. - they all look fine. Distributor side, too.
Now I'm ticked.
We set off for home, me gingerly piloting the thing home and racking my brain on what possibly could have happened...
Car ran enough to get home, but it isn't nearly right.
The snakebit syndrome continues, apparently.