FWIW, I figured out a mitigation for this: switch to desktop mode on the Deck, and then start streaming W3 from the standard (non-Big Picture) interface. You might need to move windows around if you're using an external monitor, but it's pretty straightforward to do so.

So, it's not remote play per se that is the problem: it appears to be that Big Picture mode on the client forces a controller scheme. Turning off Big Picture enables the logic for choosing the scheme based on the last HID touched (kb/mouse or controller). So I guess I don't need a mod for this, though a toggle in the Big Picture overlay to force a particular scheme would be a lot easier than having to jump through the hoops described above.