Pool shock treatment might be a bit too harsh. I'm sure it wouldn't cause immediate failure of the tank, but over time it could accelerate normal wear making it brittle and subject to leaks. A quarter cup of bleach may be just as effective and you're bound to have some already.
Part of what dish detergent does is lower the surface tension of the water. If you ever watch mosquitos, the mature ones will literally walk on water. If you have a sadistic streak (and I do hate mosquitos) just add a squirt of detergent to any stagnant water and the skeeters will start sinking like rocks. The larvae may simply lose the the integrity of their outside membrane once dish detergent enters the picture.
Carrying buckets of effluvient. This is a job for the Sewer Solution. If your plumbing is less than 60 years old, you are sure to have an "outside cleanout" somewhere on your property. Even if it is slightly UPHILL from your Airstream, the Sewer Solution, with assistance from water supplied via garden hose, can drain your tanks. It's certainly neater than the bucket route, and no tripping and spilling "IT" all over your bathroom!
Look at their website to see how it works.
Today is a gift, that's why they call it the present.