I spent some more time looking into gamepad support and found my information was inaccurate- or at least out of date! Browser gamepad APIs appear to work outside a "secure context" in most browsers today.
Decker 1.66 now includes a "gamepad" interface which abstracts over a subset of keyboard and physical gamepad input, and Dialogizer has been updated to support this feature. On a keyboard you can now navigate dialog boxes with space+arrow keys.