Is there any rule that prevents an ambidextrous pitcher from throwing with either hand to the same batter?

No.
More info on the rule
Harris, (pitched with both hands as an Expo in 1995) was with six clubs over a 15-year career. When he was with Boston, word of his odd talent got out and Bobby Brown, the American League president, gave umpires these rules: The pitcher must indicate which hand he intended to use. The pitcher may change arms on the next hitter but must indicate the arm to be used. No warm-up pitches between the change of arms and if an arm is injured, the pitcher may change arms and the umpire must be notified of the injury. The injured arm cannot be used again in that game.

addition:

earlier this year in 2010 in a minor league game there was a situation with an ambidextrous pitcher who was pitching when a switch hitter came to the plate. After about 15 minutes of the pitcher switching hands and the batter switching sides of the plate, it was determined that the pitcher must declare which hand he will pitch with, and cannot change until the next batter. It is now a rule for MLB
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