I know you can make refrigerator pickled beets with canned beets- but I don't know if you can make canned pickled beets using canned beets.

The double processing (double canning) will make them quite mushy. That is one reason why it isn't recommended - the poor quality that results. In other words, it isn't a safety issue (assuming you use approved instructions), it is a quality and efficient use of time and energy issue instead.

The approved instructions for making pickled beets is Pickled Beets

please note that it calls for a vinegar to water ratio of 2:1 rather than the 1:1 you propose. The rest of the ingredients (onions, cinnamon, cloves) are optional.

Dave