Why Ingenuity I and not Ingenuity II ?

[Faeye]

Ingenuity I costs 24 CP. Ingenuity II costs 32 CP. For raising quality, Ingenuity I and II are identical. For raising progress, Ingenuity II is better than Ingenuity I, but you can raise your base craftsmanship to the point where Ingenuity I will allow you to fill progress 100% in two Standard Syntheses, same as Ingenuity II. Ingenuity II has its uses when you're using Careful Synthesis II, but otherwise, the extra CP consumed isn't worth it. It's useful enough that you should get it eventually, but if you're just starting out crafting, it'll do to get BSM to 15 and proceed to CRP.

Ing I + 2 x Standard Synthesis = 24 + (2 x 15) = 54CP

Ing II + 3 x Careful Syntheses II = 32 +0 = 32CP

@OP
Get your Carpenter up to 50. BB is great for all kind of crafting.
Then get your BSM to 50. Ing II will save your money from forbidden melding in long run.