Open your gray tank discharge valve and see if anything comes out. If nothing comes out, try shoving either a plumbing snake or a flexible garden hose up the pipe. If something is clogging or if the valve is closed when you think it's open, you won't be able to shove the snake or the water hose up into the tank.
It's possible, you have a clog, and shoving something up the pipe into the tank from the outside might free it up. (be prepared to get a bath when it breaks free).
If the grey tank is clear, then you know the clog is somewhere between the sink and the grey tank in the drain plumbing.
Tearing the underbelly of your camper will not accomplish anything, except give you more work to do to replace it. You won't be able to see anything clogged through the black pipe anyway, and you don't want to dismantle anything as it's all glued together. Start cutting things apart and your in for some REAL work to replace it all.
If the gray tank is really empty, then proceed with a snake down the kitchen drain. Avoid drain opener chemicals. Not that they will hurt the pipes. No they will not, but the caustic affects of drain openers will not be pleasant if it get's on you, your clothing, or anywhere on your camper.
That's how I'd approach it, initially.
Question? do you put grease and garbage down the kitchen sink in your camper? Bacon grease, animal fat, beef grease, gravy, and such will clog a drain for sure. Potato peels are a disaster for drain, as are other clipping from vegetables.
Always wipe your dishes with paper towel and dispose in the trash before washing in the sink. Always absorb any grease in pots and pans with paper towel and discard in the trash before washing. These are enemies for your drains. The same is true in your stick & brick house too.
If you need cheap paper towels, go to Lowes or Home Depot and pick up a box of paper shop towels. They are inexpensive, and absorb much better than any kitchen paper towel I've ever used. We use paper shop towels for everything now. The blue in the roll or the white in a box are both great! Sanitary? They are as sanitary as any "Bounty" you'll buy.