Update: I contacted the guy who made the MAME patch that allowed for up to 22 buttons, and he was kind enough to make a custom kernel for me that made it so the adapter was recognized as two separate controllers instead of one. This was a perfect solution! It not only solves the MAME issue, but all other emulators, such as gngeo. I’m not sure if he has an account on here, but his github profile is: https://github.com/ian57 . All credit goes to him!
Since it was working perfectly, I went ahead and mounted the Sega Saturn ports in my NES case–the first step in my custom build. These ports fit almost perfectly into the NES port holes. I only had to dremel out about one mm to make them fit. If anyone is curious, the adapter is the Mayflash SS Controller Adapter for PC USB (amazon link).
I’m happy to say this issue is resolved for me. Thank you again choccyhobnob for your help! I really appreciate the efforts.