Any question answered...

...So the question: would a new lever (£20 for a pair) indeed sort this, or should I find a shim for inside the bars (a la beer can)? Since I remove the lever at the track this would not be a fit-and-forget application so a shim is a less attractive option...
I'd give the shim a go. If it works, you could glue it in place (inside the bar), thus causing less fiddlage at the track.