[Shield Q&A] Stiggosaurus:
Well, I seem to have figured it out. Seems to have had nothing to do with ePSXe, but thought I'd share exactly what I did in case anyone else runs into the same issue.

Turns out, my Shield shipped with a fairly old software version. I was rocking Android 4.1, which best I can tell was the original version that shipped with the Shield. I hadn't updated yet to 4.4 KitKat because of some of the unfavorable things I had read on here regarding the most recent release. When I realized I wasn't even on Jelly Bean, didn't have the Gamepad Mapper, and didn't have the GRID Beta, I figured it might be worth investigating getting on the most recent version of the OS prior to the most recent release. My thought was that starting ePSXe from a fresh clean slate and on a fairly recent version of Android couldn't hurt either.

Long story short, it worked great. All I really had to do in ePSXe was choose the Shield as the controller under Preferences and set the Pad type to DualShock (both of which I had done previously without any luck). Fired up Gran Turismo and it immediately saw the analog sticks!

Now getting to 4.3 Jelly Bean (Update #68 for Shield I think?) from 4.1 wasn't trivial. It wasn't difficult mind you, thanks to several great threads here that pointed me in the right direction, but it's obviously not as simple as just updating to the most recent release.

For anyone still on anything earlier than 4.3 Jelly Bean who wants to get updated, here's exactly what I did:

DO NOT update to 4.4! There's no need to do this--you can jump right to 4.3 from 4.1.

Enable USB Debugging.

Install the Universal Naked Driver using the instructions here.

Download the Shield RAM 1.0 utility here.

Launch the Shield RAM utility and perform the Driver Check (option 1). Note that you must enable USB Debugging and have installed the Universal Naked Driver indicated above in order for this test to pass. You must make sure it passes both the ADB driver test AND the fastboot driver test. I had to end up manually installing the driver for Fastboot, so be aware this may be something you have to do as well.

From the Shield RAM utility, choose option 2 to unlock the bootloader. NOTE! This will technically void your warranty AND wipe your device. Apparently Nvidia is still pretty cool about replacing devices with unlocked bootloaders, but just be aware that the warranty technically will be voided if you proceed.

After the bootloader is unlocked, your device will be restored to factory defaults (still on 4.1 at this point). When the device reboots, run through the initial setup and then go re-enable USB Debugging.

From the Shield RAM utility, choose option 6 for "Reflash Bone Stock". The stock version included with Shield RAM 1.0 is 4.3 Update#68, which I believe is the most recent release prior to 4.4 KitKat.

The stock images will be flashed to the device, and once complete and device reboots, you'll be running 4.3, Update#68! This release includes Game Mapper and Nvidia Grid Beta.

Ideally there'd be a way to get to 4.3 without having to unlock the bootloader and void the warranty, but I couldn't seem to find anything that indicated this was possible. This is probably pretty basic simple information and instructions to a lot of people, but it's exactly the process I went through to get on 4.3 AND fix ePSXe's analog stick issue at the same time. Hopefully someone finds this useful! :)