OK, I would select the brim in the materials list, make the diffuse colour to match whatever the rest of the hat colour is. change the Transparency value to 1.0 and also the Transparency edge to 1.0, and make sure Trans Fall off is 0. Plug the lace tile into the transparency node. You will probably have to change the U_scale and V_scale of the trans map, depending on how big the tile is (I mean, how often it repeats, or if it is just one tile.) Does that help?

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"you are terrifying
and strange and beautiful
something not everyone knows how to love." - Warsan Shire