[Arran]

Finding proper veneer isn't a serious issue, from what I can figure out, using a straight edge and an automotive type feeler gauge, most of my radios seem to have used veneer 1/52 of an inch and thinner. The paper backed veneer is so ridiculously thin that I have never contemplated using it for anything other then replacing a faux finish like "Philcote", and unlike the solid veneer you also have a much reduced selection of cuts and species to choose from.
If you want to cut back on the quantity of tone spray that you have to use you could always dye the wood underneath it first, I have had a habit of using brown fabric dye like Rit or Tintex to darken it up first, aniline wood dye is another way. Under a colour like extra dark walnut tone spray it will obliterate any lighter shade of brown anyhow.
Best Regards
Arran