I plan to competitively drive my 2008 X3 over ice courses. Would the addition of a Dinan (or other) Tower Strut Brace significantly improve handling on ice and mountain roads?

BMW's are pretty stiff already, but it couldn't hurt.

Ice racing sounds like fun! Not much chance here in Florida.

Your X3 already has a nice solid (not hinged) strut brace, the best kind. I can't imagine a different one would improve handling. Your best gains would probably come from lowering springs (e.g., H&R). Assuming a course without huge moguls, a lower center of gravity should give you more responsive steering, less body roll, and faster times. The H&R springs have a minimal drop, maybe 3/4 inch, but that makes a huge difference in handling. Since they're progressive, they also soak up small road imperfections in normal driving better than the stock springs.