I decided to go ahead and buy a Xiaomi QIN 1s feature phone. Unlike other feature phones, it runs a version of Android. Since there doesn't seem to be any threads here on XDA, I thought I'd start one.
I finally found a guide/tool over on 4pda that will let you install apks. I have translated the tool to English and have it attached. The original post of the tool is here
If you don't want to download the tool (which also helps with ADB) here are the steps to obtain root:
-- Install the driver on your PC
-- Dial: *#*#83781#*#* To Open the Engineering Menu
-- Find the section "Debug & Log" and enable "Allow Debug"
-- Open the Settings app
-- Find "about phone"
-- Click "Build number" multiple times until you see "You are now a Developer!"
-- Go to "Developer options" and then turn on USB Debugging
-- Connect your phone to your computer (MTP)
-- Using ADB, push the apk to the device
You can use this method to install any applications, not just root the phone. Remember, however, we only have 256mb of RAM and a total of 512mb of storage. Definitely not for gaming and you will definitely want an SD Card. I can confirm QQ and WeChat work fine when installed.
Let's get this rooted!
Big shoutout to Frolkin who helped me get the original tool and gaguga for making the tool!