After checking back on my substance I realize that this solution unfortunately doesn't apply.
For example: I'm using the built in brick generator and want to randomize the number of bricks so that they can be 1x1, 2x2, 3x3, or 4x4. If I randomly choose an appropriate Integer2 to pass to the brick_generator graph, then the glitchy output happens. This is really surprising to me as I would think scoping would prevent new randoms from being provided by the parent graph.
If I instead generate the random values right inside the FX-Map's iterate node (inside the Tile graph), then the brick_generator graph has no knowledge of what value was chosen there and many of the other moving parts will fail.
Is there really no way to pass a randomly generated parameter to a graph and have that graph run to completion with the value you passed it?