Bigfoot carpeted ceilings -clean, remove, cover?

Our AF has a white interior ceiling. When we bought it, previous owner had not clean the AC filter regularly. I'm guessing they did a lot of cooking at the lake which left greasy black/dirty areas out about 12" in front of the both AC vents.
I used a brush and carpet stain remover. I probably did it 6 or 8 times but it finally removed it all. Can't tell it was ever in a mess like that.

I think that steam cleaner will also work great. Personally I don't think that this a problem. I always left the room with clean carpet for a day and the smell disappeared. I often used dry cleaning of carpet and I have never had any problem. This dry cleaning is not harmful for pets or children. The fact that they are the ones who get my carpets dirty and I often use dry cleaning. I am very glad that now there is such a miracle of the 21th century, where I can just take the carpet to the dry cleaners and have it perfectly cleaned. I've been contacting this company all the time lately sunlightfinerugcare.com, because they do their job good.

1 look (google)up marine head liner !!
2 purchase "insta gone " it will remove your stains
3. re seal ac

I wouldn't be in a big hurry to steam clean. Material is glued to Luan with contact cement.
On my 2002 BF I washed the roof one time without having a vent completely shut by accident and shot water down the vent. It caused a lot of brown staining a couple of inches out from vent trim on inside ceiling. I'm pretty sure the brown stains bled thru from the wet Luan wood...
I left it for months trying to figure out what I should do, then wife said maybe try using Oxi-Clean laundry stain remover on it. I pulled the plastic trim off then used an Oxi-Clean spray bottle to wet material, then right away used a small sponge to work it so I wouldn't pull loops out of the fuzzy material like a brush might. I had my wet/dry shop vac there too and sucked the cleaner out of the material after working it. I went over it twice and later when it dried completely there was no trace of a stain or that the area of fuzzy stuff had been worked on...only thing I would be careful about is if the whole ceiling is too dirty then the cleaned area might show as a whiter patch.

I bought our camper from a rental fleet. When I went to pick it up they loaded it on the truck and took it inside the shop. Their cleaning lady then spent 5 hours cleaning the camper, while the mechanics installed the tie downs, etc. She had a commercial grade steam cleaner and the entire ceiling felt slightly damp afterwards, so I assume she steam cleaned the ceiling as there isn't a lot of anything else to steam clean.
I would take it to a reputable carpet cleaner and let them handle it.I have had a bad experience trying to use a rented steam cleaner on a too dirty carpet that never looked right even after 2 steam cleanings.
The carpet is glued to a thin plywood, which is glued to the white bead board insulation, so removal and install of something different might be a challenge.
Good luck with your decision.

I will use 1/4 white vinegar to 3/4 warm water. Put it on a white towel and blot the areas. you can spray it on but if its like car headliner you don't want to get it to wet.

I clean carpets and do flood work and sometimes have to take browning off of carpet from water damage and we use Fiber Rinse which is an acid and so is white vinegar. What you are trying to do is neutralize alkalinity and residue.
Hope that helps. If not you can call any carpet cleaning supply store and ask them to direct you to the right chemical but I would try the white vinegar first.

Mold and Mildew Stain Remover by Zep works well but it has a strong odor that takes a few days to dissipate after use

Steam cleaner? Talk to a car detailer, they may have an idea.