DualShock 2 Controller No Longer Working under Proton 5.13

Since the introduction of Proton I've been using a DualShock 2 controller (connected via a USB adapter) without issue, but noticed about a month ago that it is no longer being detected by most games running through the latest versions of Proton. It still functions perfectly in Big Picture Mode menus, where it shows up as a DualShock 3 controller, but gets no response in-game.

So far I've tested this with Dark Souls: Prepare to Die Edition, Dark Souls II: Scholar of the First Sin, Hades, and Sonic Adventure 2, all of which no longer respond to input. I've also tested it with Yakuza Kiwami, which (to my surprise) does still respond to controller input. Forcing Steam to use Proton 5.0-10 restores controller functionality in all games listed. Native games still function as normal (tested with Baba Is You).

So far I've tried:

• Switching to Proton 5.0-10 (works, but not an ideal solution)
• Disabling Steam Input (no change)
• Using native runtime (no change)
• Opting out of Steam client beta (no change)

Logs produced by Hades:
5.0-10 (working): https://gist.github.com/PolyCement/2514f03aed918cacc4743ec9c6870899
5.13 experimental (not working): https://gist.github.com/PolyCement/0128fad86e91a01304c5b3071e1b442c

System information:
https://gist.github.com/PolyCement/a1b5d939ab67622929e333a4701fbe59

Start by comparing the linked Hades logs for Proton 5.0-10 and 5.13 experimental, then review the linked system information and reproduce the issue with the listed games. Done means DualShock 2 input works in the affected games under Proton 5.13 without regressing Yakuza Kiwami, native games, or Proton 5.0-10.