Does anyone have a feel for how heavy the weight of an equivalent oil(?) driven ship would be? It has the big number for the weight of batteries, but I've got nothing to compare against.

700-ish tonnes - it's in the wikipedia article:

"In 2020, Buquebus originally commissioned Incat to deliver a new ship to use dual-fuel propulsion, capable of operating on liquefied natural gas and diesel, with around 400 tonne of main engines, 100 tonne gearboxes, 100 tonne cryogenic fuel tanks and 100 tonne fuel."

https://en.wikipedia.org/wiki/China_Zorrilla_(ship)

Google supplied me with this original spec: https://incat.com.au/wp-content/uploads/2019/05/Mini-Spec-13...

Same ship, originally specced to carry tens of thousands of liters of oil. No overall displacement number, oddly.