So it's the Backspace key that is the culprit in Wine/Prox1. Delete key didn't work yesterday because I had chosen, in Keyboard layouts, to make it a Backspace key. Once I disabled that, the Delete performed with a single press and with no subsequent doubling of characters/numbers.
Googling has found similar complaints in other Wine setups, but here, I'm thinking that it's because of the hardcoded Pro1x ability to swap the Delete key for a Backspace.
But what do I know, having turned my back on Dry January 🙂