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Rented gear puller bolts don't fit the threads in the harmonic balancer. 5/16 is too small and 3/8 is too big. Same with 8 mm. Haven't found anything in between.
3/8" should have worked. You probably have years of built up gunk that is preventing the bolts from fitting.
A three jaw puller is generally not recommended, since the pulley can be easily damaged.