The core design problem with using the Patriot missile system for ABM applications is that its terminal guidance system fundamentally isn't designed for it. It uses radar for terminal guidance because it was adapted from anti-aircraft applications. That this works at all is impressive, in a "dancing bear" sense, you wouldn't expect the discrimination to be good enough for the purpose.

All purpose-built ABM systems use infrared imagers for terminal guidance with classifiers that pick out both the target and desired point of impact on the target. These are vastly more effective but we also don't sell them to foreign countries AFAIK.