Why don’t you apply a thin little black stroke (layer effect) outside the white text so that it’s still visible on a white background ?
I could probably work out how to do that in Photoshop but someone who does embroidery should be able to use this directly. They would not want the background because they don’t want to end up stitching (or printing) the background.
Another thing is that the white and blue may not show up on some backgrounds. Obviously they won’t work for a white or a blue t-shirt. In that case the shop will need to change the colours. For a black t-shirt the logo will probably work fine.
I was thinking to somethiing like that.
The stroke is the blue reference you used for the GA and the ‘nike’ curve, and on a white background, the “euro” and “.org” text is still visible, so you don’t have to pay attention to the t-shirt’s color.
PS: I don’t know which version PS you have, but to create the stroke effect, select your text layer, right click on it and go to blending options..
Hope this helps :-)
or another version with a white outline on the GA and ‘nike’ curve… It’ll fit with any t-shirt colors….
Shadows and 3D are out, flat design is in 
