I used ExpertFlyer to pull up fare rules of the cheapeset QR one-way coach fare FRA-SYD. As I understand it, most or all QR fares should be identical in this respect.
The fare provides in relevant part (bold by me):
IF THE STOPOVER IN DOHA BETWEEN CONNECTING FLIGHTS
IS LESS THAN 24 HOURS. PLEASE NOTE THAT THE
PASSENGERS HOLDING A NEW TRANSIT VISA HAS THE
OPTION OF EITHER CHECK-IN THEIR BAGS UNTIL THEIR
FINAL DESTINATION OR ALTERNATIVELY UP TO DOHA
HOWEVER IF PASSENGER DOES NOT HOLD TRANSIT VISA
THEN BAGS WILL ALWAYS BE CHECKED-IN THROUGH TO THE
FINAL DESTINATION. THE PASSENGERS
SHOULD CARRY EVERYTHING THEY NEED IN DOHA IN A
HAND BAG
With your 20 hour stopover in Doha, and leaving the airport, I expect that you'll get a transit visa. Get that, bring proof of that, plus these fare rules -- and you should be able to check bags only through to Doha. With this document in hand, don't take no for an answer!
I've found that QR staff were previously trained to be quite strict -- harsh! -- in enforcing the carrier's prior rule that bags must be checked through for all connections <24 hours. Some may not know the above rule. But that is the rule in the fare, and you are entitled to its benefit.
(Ideally of course you'd confirm the exact provision in your fare rules, based on the ticket you bought and the fare in that ticket, on your date of purchase. EF lets you check historic fares and fare rules.)