Do you have any components that would allow me to build a two axis tilt (θ-x, θ-y) platform without any screws protruding up above the surface?

Edmund Optics® offer Metric Goniometers which can be stacked to give two-axis tilt. Our TECHSPEC® Kinematic Table Platforms also have built-in tip/tilt capability. Both of the above the two-axis movement you require without any screws to interfere with your setup.