As others have mentioned, one day isn't long for this sort of caper, given all that's involved in getting there and back.
An alternative, which offers a somewhat similar experience, is to pick up a 4WD at Tin Can Bay and drive out to Double Island Point, past the wreck of the Cherry Venture, etc. Obviously you're not "on Frazer Island", but the geography is pretty much identical and if you look at a map it's apparent why. Big advantage is no ferry transfers, etc -- just drive in a 2WD from Brisbane to Tin Can Bay, pick up the 4WD, straight onto the beach and off you go. Reverse the process at the end. Much more efficient when time is limited.