How can I rebind the return key so that it works consistently across all applications?
You probably need to step a level further down the tree of software and hardware, to the level of KyRemap4MacBook - which acts as a filter between the physical keyboard and the keyboard events reported to MacOS.

The software keyboard map is an optional thing - software can intercept keyboard events at a level that bypasses it, if they wish, and it sounds like some of what the browser does with key binding is touching on that.

A lower level remapping may bypass that problem.