Un-moving "static" image from VGA controller
Short answer:

The SRAM "chip enable" (/CE) and "output enable" (/OE) pins were not properly driven, and that caused the error seen in the original post.

Long answer:

After some back-and-forth in the comments with Dave Tweed, it was determined that the SRAM, which acts as the VGA frame buffer, was not being properly initialized. The video output from the DAC was simply uninitialized information that existed in the SRAM. Otherwise the FPGA and video DAC seemed to be operating correctly, allowing us to narrow down the issue to the SRAM.

After a brief review of the SRAM connections in the schematic, we found a few mistakes and missing connections:


There was no connection to /CE1 (chip enable 1). Therefore, the chip was not being properly enabled.
/CE3 (chip enable 3) was improperly labeled as /OE3, and not connected to /CE1





The output enable pin of the SRAM, /OE, had no driving source. It was only connected to the input of the NC7S04 inverter. 




The solutions were simple: 


Connect /CE3 to /CE1





Add the missing connection between /OE and the FPGA (discovered to be Pin #143, "IO_L10N_0/HSWAP"), which has a 4.53k pullup to 3.3V




I also discovered that the ZZ (sleep enable) pin of the SRAM on the original board was tied to the MODE pin, which was pulled to ground through a 4.53k resistor. This was not as crucial, though, because the chip already has an internal pull-down for the ZZ pin. Nonetheless, I marked it as a required change.

I made the modifications by soldering some #34 AWG enameled wire to the appropriate pins on the clone board and plugged it into my scope. Sure enough, everything worked as expected. I guess only 2-3 missing connections on a reverse-engineered 6-layer board that was completed without damaging the original is pretty good, all things considered!