I live in Sydney, and we go up to the Whitsundays at least every year if not more often.
Here are some websites of companies operating in the region.
Bareboat Whitsundays, Cumberland Charter Yachts, Airlie Beach, Whitsundays, Australia ( cumberland charter yachts)
Whitsunday Escape
Sunsail Yacht Charters Asia Pacific : Luxury Yacht Charters
Whitsunday Bareboat Yacht Charters - Sailing Whitsundays
All have cats. There are other companies in the region as well but the above I have all had firsthand experience with and would happily recommend.
Any other questions, please feel free to ask.