An easy method might be to use the antifreeze injection port - if you have one.
On my coach, there is a tube designed to inject pink RV antifreeze into the water line. It is located between the tank and water pump and connected to the water line via "tee" fitting.
There is also a valve in the water line between the water tank and pump/antifreeze port that becomes part of this system.
If the valve is off, it allows the RV's water pump to suck antifreeze into the water lines via a jug with the stuff attached to the antifreeze port. But if the valve is on (and the pump off), since the antifreeze port is attached to the exit line of the water tank via Tee fitting, you should be able to gravity feed sanitizer into the tank.
The blue tube in the photo is the antifreeze injection point for my water system.
If yours is similar, you may be able to dump sanitizer into the tank directly from the antifreeze port with the valve on and water pump off.
My suggestion is to make an adapter to attach the end of the blue tube (it is a 1/2" NPT for my coach) to a bucket, and pour the bleach solution in the bucket, make sure the antifreeze valve is on, and leave the pump off. It should work.
You might also have to remove the fill tube cap from the outside of your coach so you get enough ventilation into the tank that the solution in the bucket will easily fill the tank.
Another idea I have is to make a "bucket" by purchasing a Camco 66501 boat winterization kit - which basically consists of a plastic jug (with vent in the top) a hose, and a valve. The end of the hose is a garden hose, but you can either cut the end off and install a fitting for the antifreeze injection tube, or perhaps finding a female garden hose to male 1/2" NPT adapter.
Then, simply put the solution into the plastic jug and attach to the antifreeze injection port. Make sure the water pump is off and the water line valve is in the normal position, and let gravity do the work.
I'm going to make one of those jug things for next season.
I like the idea of re-purposing the boat winterization kit, as it has all of that is necessary to do this without spilling stuff all over and making a mess.