Determining Blanking for HDMI Transmitter
After some extremely helpful discussion in chat with Justme, I believe I have found the solution.

I began by probing points around the VGA reference board, which I am converting to HDMI (or at least DVI-D, I'll need to look into this per the comments on the original post). I measured Vsync, Hsync, and the pixel clock. I also managed to solder a very fine wire on the blank pin of the original VGA driver chip (which, like an idiot, I overlooked in my design) and was able to measure the blanking interval. I came up with the following:


Vsync = 60.125 Hz
Hsync = 37.879 kHz
Pxclk = 40 MHz
Blank = 6.4 us
EN = 20 us


I also decided to simply plug in the reference board to a monitor and view the monitor's settings after it auto-configured. I'm not sure why I didn't do this from the beginning. The monitor showed a resolution of 800 x 600 px rather than the 800 x 480 that I thought, which was only the native resolution of the device (not the external monitor resolution). I found this webpage showing the timing of an 800p x 600p x 60 Hz VGA signal which was consistent with my measurements:



Now that we had determined the correct resolution and timings, I was able to determine the correct register values. Just a reminder, these were the registers for which I needed numbers:


DE_CNT - display width (in pixels)
DE_LIN - display height (in pixels)
DE_DLY - horizontal sync pulse + back porch (in pixel clocks)
DE_TOP - vertical sync pulse + back porch (in pixel clocks)


Using the table of timings on the tinyvga website, I was able to determine the following:


DE_CNT = 800 pixels = 0x320
DE_LIN = 600 pixels = 0x258
DE_DLY = 216 pixels = 0xD8
DE_TOP = 27 pixels = 0x1B