Chris, Sipan island https://www.croatiaferries.com/sipan-island-ferries.htm is about one hour away from Dubrovnik.
You can go there by three ferries, all departing from Dubrovnik, among which just one takes cars and scooters, the one that sails to Sudurad port:
https://www.croatiaferries.com/sudurad-ferry-port.htm
That is car ferry line 831 served at the moment by Mate Balota car ferry – see video
If you rent a bicycle you can get it on any of three ferries – two sails to Sudurad and one to Sipanska Luka, which is a further point on the island:
https://www.croatiaferries.com/sipanskaluka-ferry-port.htm
Re: I understand this is a very small island and I’m not sure if this would be a reasonable option.
Yes, Sipan is a very small island and has hardly any roads, actually, it has just one main road that is connecting Sudurad, where the car ferry arrives, and Sipanska Luka, which is about 4.5 km distance. It is easy to walk this distance or cycle and I don’t think it is worth taking a car there. Cycling should be fun too. And for the scooter – yes, that is also an option, just make sure you rent a scooter that is sturdy enough if you intend to go on side/country lanes.
Here is a driving map:
