I got sound working in Turrican 2!
After some more troubleshooting, I discovered what the problem was.
Just prior to testing Turrican 2, I had decided to clean up and organize my AUTOEXEC.BAT file. The GUS Extreme setup program had inserted lines into the AUTOEXEC.BAT file including the @ symbol in front of each line to prevent displaying each line during boot up. Since I use "@ECHO OFF" at the beginning of my AUTOEXEC file, those @ symbols seemed superfluous and I removed them.
One of the lines used by the GUS Exteme's initialization is setting the BLASTER environment variable. Removing the @ symbol in front of that SET BLASTER line caused the CTCM driver used by the AWE64 to overwrite that line. It looks like the CTCM driver scans the AUTOEXEC.BAT file automatically and replace the SET BLASTER line with whatever the settings are used in the CTCM config.
This resulted in the GUS Extreme not initializing because it was trying to use the same settings for the ESS1688F chip as was assigned to the AWE64.
Using those settings in Turrican 2 resulted in no sound being played.
After noticing this I replaced the SET BLASTER environment variable with the prior settings for the GUS Extreme's ESS1688F chip including the "@" symbol. This prevented the CTCM initialization from overwriting this variable, thus allowing the GUS Extreme to properly initialize.
I retested Turrican 2 choosing SB 16 and using the corresponding settings for the AWE64 and it worked! I also retested it using the Gravis Ultrasound and that too worked, albeit with some audio issues likely owing to the lack of 100% compatibility which is a known limitation with that game anyway. However, I still don't have an option to use the settings for the ESS1688F chip on the GUS Extreme, since I'm using address 230h which it won't let me select.
I also double checked other games and everything else still seems to be working.