I was able to solve this after hours of tinkering. I'll give the process I followed below. For reference, here were my settings for the previous BIOS versions and current BIOS version (1203), so you can see how they have changed.
Motherboard: ASUS PRIME Z270-A
CPU: Intel Core i7-7700K Kaby Lake Quad-Core 4.2 GHz
Memory: G.SKILL TridentZ DDR4-3200 16GBx2 CL16-16-16-36
BIOS 0906 (XMP works)
XMP: Enabled 16-16-16-36 @ 3200Mhz
DRAM Voltage: 1.3530 (XMP specification value configured by XMP profile)
CPU VCCIO Voltage: Auto
CPU System Agent Voltage: Auto
BIOS 1009 (XMP does not work)
XMP: Enabled 16-16-16-36 @ 3200Mhz
DRAM Voltage: 1.3728 (Configured manually)
CPU VCCIO Voltage: Auto
CPU System Agent Voltage: Auto
BIOS 1203 (XMP does not work)
XMP: Enabled 16-16-16-36 @ 3200Mhz
DRAM Voltage: 1.3530 (XMP specification value configured by XMP profile)
CPU VCCIO Voltage: 1.1375 (Configured manually)
CPU System Agent Voltage: 1.1375 (Configured manually)
Sorry if there's a lot of noobish stuff in here. I'm a noob. Big whoop, wanna fight about it?
The first thing I tried was what I did last time (for BIOS 1009), which was increase the DRAM voltage. Once I started getting into the 1.4v area, I got worried it was too high and stopped. I decided to try getting it stable at a lower frequency to see what the automatic settings look like...
I enabled XMP and manually adjusted the DRAM Frequency to a value lower than the XMP profile's 3200MHz, something like 2933MHz. I picked lower frequencies until I was able to get the PC to boot, and more importantly, to get the BIOS settings to stick (the settings are reset to default if it fails to boot). Setting it to one of the DRAM frequencies higher than the stock 2133MHz allowed me to see what sort of DRAM, VCCIO, and System Agent voltages the system ran with. I saw they were significantly higher than usual, like DRAM voltage of 1.4720, VCCIO 1.024, and System Agent 1.232.
Based on the values I saw for VCCIO and System Agent voltages, I realized I wouldn't reach stable automatic values until I move the DRAM voltage much higher for 3200MHz. After some quick reading online, I realized I could safely (temporarily) increase the DRAM voltage to around 1.5, so I set on a new path...
I kept XMP enabled and switched the DRAM frequency back to the XMP profile's default of 3200MHz. I then incremented the DRAM voltage starting at the XMP spec value of 1.3530 (leaving VCCIO and System Agent on Auto), and restarting after each increment. I did this knowing I would probably have to surpass the 1.4720v required for the lower DRAM frequency, but I wanted to see what happened with each increment. When I finally hit DRAM voltage of 1.5048 (yes, it's high for this RAM), the system booted.
I took note of what the BIOS interface displayed for the current values of the VCCIO and System Agent voltages, which were still set to Auto. They were:
DRAM Voltage: 1.5048 (Configured manually)
CPU VCCIO Voltage: 1.2560 (Auto)
CPU System Agent Voltage: 1.3360 (Auto)
These values are all pretty high.
At this point, I turned the DRAM voltage back down to the XMP spec value of 1.3530. The VCCIO and System Agent voltages are functions of the DRAM voltage, so at this DRAM voltage they would be too low if set automatically. I configured the VCCIO and System Agent voltages manually to values as close to the automatically set values when the DRAM voltage was 1.5048, so it looked like this:
DRAM Voltage: 1.3530 (XMP specification value configured by XMP profile)
CPU VCCIO Voltage: 1.2625 (Configured manually)
CPU System Agent Voltage: 1.3760 (Configured manually)
This configuration worked. I was able to boot with the XMP spec DRAM voltage of 1.3530. The VCCIO and System agent voltages were still high though. I decided to try decrementing them and was surprisingly able to get them as low as 1.1375v, which is the value just below when they start to go above spec (turn "yellow").
My final settings look like this:
XMP: Enabled 16-16-16-36 @ 3200Mhz
DRAM Voltage: 1.3530 (XMP specification value configured by XMP profile)
CPU VCCIO Voltage: 1.1375 (Configured manually)
CPU System Agent Voltage: 1.1375 (Configured manually)
I hope this helps someone else.