I found a (Japanese-language) report about the clock mechanism, with some interesting photos. From context, I'm guessing this is a product of the 2004 reverse-engineering program described in OP. For example, page 4 has a photo of the (very weird) gear OP describes as "rhinoceros beetle" shaped, with the modern replicas next to it.

[.pdf] https://www.global.toshiba/content/dam/toshiba/migration/cor... ("万年時計の機構解明 Mechanism of “Man-nen dokei,” a Historic Perpetual Chronometer")

(I'm fairly sure no English translation of this document exists anywhere, at least indexed on Google).