Problems burning bootloader on custom ATMEGA2560 board for High-powered model rocketry
I have thus far been completely unable to burn the bootloader.


In what sense? What (error) messages did you get?

In your shoes I would be seeing if I can "talk" to the chip. My chip detector sketch will help you to see if the chip is "alive" or not. See my post about Have I bricked my Arduino Uno? Problems with uploading to board

You may possibly have inappropriate fuse settings for your clock source. If you get some sort of response from the chip using the chip detector sketch please post what it is.




  Also to note is that this is a completely virgin chip from the manufacturer, so I'm not sure what the clock settings come as in that case


According to the datasheet, 8 MHz internal oscillator with CKDIV8 fuse set. That should work.

However you need to resolve that before going any further. I would be taking voltage readings at various points. 


Make sure you have Vcc at all the appropriate places, and Gnd.
Note that there are multiple Gnd and Vcc pins. You should also have 5V at AVcc. 
Check that /RESET is normally high (5V) and check that it drops to 0V during the programming attempt (and stays there).
Check you have decoupling capacitors between Vcc and Gnd where necessary.
Assuming you have the TQFP-pinout chip, make sure that you have your SPI connections at pins 19, 20, 21 and 22.
If you have a scope or logic analyser make sure that there is some "action" on SCK, MOSI pins during attempts to program it. There should also be a response on MISO.
There should not be a capacitor in series with /RESET as I have occasionally seen people do. The programmer has to hold /RESET low during the entire programming sequence.