Brygun, I did some extra testing regarding whether or not the adze axe was being asked as the preferred axe for the punt-making steps, and it doesn't seem like it works at all for custom tools. Here's the code I was using to test:
.Punt. [effort:0] [phys:hands] *COMMON* /1/ [noquality]
{Rock} [noquality]
//{Knife} <Draw knife>
//{Axe} <Adze axe>
{Draw knife} <Hunting knife>
Neither the commented-out knife nor the axe asked for any preferable tool. By contrast, the last line did mention the hunting knife as a preferred tool but decided to use the draw knife in my inventory in spite of the masterwork hunting knife being carried as well. {Cutting weapon} rather than knife or axe also didn't work. It seems that custom tools are just not accepted in that field?
This is unfortunate, since it means we'll have to force the use of those tools or otherwise ask the player to roleplay it. The effect of using an inadequate tool is also lost.