Since you said it's your headliner I can't imagine your bedroom getting so hot from an exhaust leak that would cause your headliner to droop.
In a lot of Winnebagos with foam backed vinyl headliners, they will droop in places; and the cure is not always ideal.
You need access to spray contact cement; and that means you may need to pull down a good corner of your head liner to repair a drooping area.
In you bedroom you have a center ceiling light that helps to hold the headliner. However, in other parts of the coach there is no such extra support.
So some people choose to add a ceiling strip of wood (see pictures) or they may pin up "star shapes" or anything they find acceptable that will help hold the headliner to the 1/8" luan (wood panel)... after they use spray contact cement.
This is the problem is that 3M 70 or 80 and even their 3M headliner spray will melt the foam and that cause the soft vinyl to look terrible when you glue it back up.
The same thing will happen with most other types of contact spray, but I found the Loctite 300 is the best one of the bunch. You can buy it at Walmart or Amazon.
Be sure to use a lot of it on the foam and soft vinyl; and be sure your contact cement is good an dry before you attached the 2 sprayed pieces together. A rolled up towel will help you smooth the soft vinyl out... and don't be surprised if your soft vinyl stretches a little bit.
NOTICE TO ALL: Never clean your ceiling headliner if you can avoid it! The pressure breaks the glue bond and within months of using vinyl cleaner your ceiling headliner will start to fall.
Winnebago charges $1500-$2000 per panel and they will not do spot repairs. In addition, you may be hard-pressed to find anyone to do this repair, because most never turn out very attractive. And if you go with new panels in some areas you need to know the new vinyl material will not match your old vinyl panel next to the repaired panel.
Attached are some ceiling repair examples.