Just wanted to update in case anyone else has a similar setup to me.
I got the BIOS chip in the mail. It has version 2302 of the BIOS on it. I bought it from BIOS Depot, which is an eBay seller. Once I replaced the chip with the newer version (3602?) with the older one, audio finally worked. I had to remove any "Inject" values that I previously had in Clover Configurator, and reinstall AppleALC with Multibeast. Then audio was working fully.
Then I tackled the issue of no hardware acceleration with my aging 560 TI. I bought a Radeon HD 7970 on Craigslist for $80. It's not the Mac version. I put this card in, but it wouldn't boot with it for some reason. I suspect it was because I had Lilu and WhateverGreen installed on the old installation, which is overkill for this card (since it should work natively).
I then tried to boot from my USB installer that I used to install Mac OS previously. It wouldn't boot with my motherboard. This is a common issue with the older version of this BIOS. So-- I swapped out the 2302 chip with the 3602 chip, booted into the USB installer, and redid the installer. This time I kept everything perfectly default. The installers all had a 1440p resolution, so it was definitely using my GPU properly. Once I got booted into the full installation, I installed Multibeast and used it to install AppleALC and the Legacy bootloader. I rebooted from the hard drive-- it didn't succeed. So, I booted back into the USB drive, selected my Mojave installation on the hard drive, and booted back into that. This time I ran Multibeast and installed the UEFI bootloader. Rebooting and removing the USB drive resulted in a successful boot. The GPU worked OOB-- no kexts or modifications of any kind.
I then had to tackle audio; it doesn't work in 3602. So, I turned off the computer, replaced the BIOS chip with the one with BIOS version 2302 on it, and booted it back up. This time, using the default AppleALC that Multibeast installs and no other modifications, audio works. Audio also works through the HDMI port on my 7970.
The very last thing that I need to fix is USB3. I couldn't ever get it to work, but at this point, I'm burnt out with diagnosing and fixing MacOS issues.