Reinforcement falls under another Magic Attribute (i.e. a different application of the elements from Projection). In other words, Reinforcement and Projection are not intrinsically related.
Shirou's version of Projection relies on UBW, so if you take away UBW, he cannot project, but he might still able to reinforce.
EDIT:
This is probably also why Shirou can't "reinforce" Ass Dagger into the Jeweled Sword.