First of all, if you absolutely need to keep the background a separate sprite, add identical alpha and slicing setup to it. Computing actual clipping to add it behind the frame is an overkill.
Now, for the tiled texture overlay, yeah, that's not possible using exclusively existing UI elements. There are two ways out of this:
A: Rethink the design to remove the need for tiled areas in all border quads, then use Advanced sprite type that allows you to set fill type individually on per-quad basis and use Tiled type for the central quad. Or maybe create an inward transparency fade in the sliced sprite, disable the fill of the central quad and insert the anchored tiled sprite underneath, that works too and would allow you to get the tiled pattern closer to corners (although you'll still have to use the inward halo with flat color).
B: Use the approach practically every single game with heavy effects on UI elements does: post processing through shaders. Create a shader and a component following the Image Effect pattern in Unity that will overlay your UI camera with scrolling texture masked using alpha of UI camera output.
I'd recommend the last one, it's clean object hierarchy-wise, it gives you enormous freedom to do stuff like glitches, varied scrolling control etc at a completely flat cost.