Putting Black Tank Chems in Grey Tank?

Use chemicals as you want......

Personally I remove 'P' traps, take them outside and scrub them clean of any gunk, reinstall and fill back up with water.....water is a vapor barrier

Then I check the Air Admittance Valve(s) under sinks. If they aren't closing properly grey tank odors can come back inside rig.

X2 on the P-trap stopping the tank odor from coming into the room.
You might try to see if there is a buildup of crud below the sink and above the P-trap. I had that in our bathroom sink and had to bust out the plastic that kept stuff from falling into the drain so I could get a rag down there to clean the piping.

We use bleach or apple cider vinegar to clean grey tanks.

Bleach or soap..hot water
I never put blue stuff, in the grey tank
One of the enzyme , or other bio friendly cleaners should be ok in the grey tank

We've had to do it in higher temps. Works fine and wont hurt the plumbing and rubber seals like more caustic stuff can.

As said, it won't hurt a thing. Both tanks and piping are made from the same materials. It's what goes into them that makes the difference.

Just use ordinary bleach.

we use 1 cup of Dawn Platinum dish detergent and one cup of Borax powder dissolved in hot water. No odor.

Won't hurt a thing.
One of the things I try to remember to do after emptying the grey tank before heading home, depending on your unit you have 6 to 10 gallons of hot water in your water heater. Drain the grey tank. Close the drain. Now go inside and run all the hot water you have into your sink drain. Let it sit for 5 minutes and then drain the grey tank again. Your grey tank catches a good bit of grease from cleaning up after dinner. The hot water can help soften up those greases and get them moving out of the tank. After the second draining add some more water and chemicals to leave in the tank. It will slosh around on the way home.

X-2 no problem