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Well, I'd say regression fitting's stability would not be the real important problem with it (though yes that would be an issue with some types of regression). The bigger problem would be that even in the impossibly best case scenario (taking into account all variables, and practically infinite data), you would still end up averaging the movement for any given scenario. Many robots are either randomized or intentionally avoid reacting the same way to a situation as they did in the past. Due to this, the average result for a given set of inputs, frequently does not match the most common result.

For exactly that reason, one feature common to nearly all robocode targeting algorithms that are stronger than simple pattern matching, is that rather than outputting a single firing angle, they output a histogram of probability of how likely different angles are where the enemy is expected to be. (Sorry about the run-on sentence, tis late at night here and my brain doesn't word things nicely at this time)

It's interesting to hear about this competition of coworkers. Have fun! :)