The park cyclone tool works well for me. I use it with mineral spirits. Take off rear wheel an install old hub to keep chain up.
Lifu also carry a tool that keeps the chain under tension should anyone need it.
In case you don't have a spare hub and are still on speaking terms with your LBS [kidding], one of those plastic thingies the manufacturers use to protect the fork and rear train from bending inwards works great too for hanging the chain on.
They usually just flip the things into the dustbin, so why not ask for a set?
I always keep some handy for that and whenever the bike needs to go inside the back compartment of the car for travelling.
Either way it looks as if the rear wheel needs to be removed to access the other side of the pulleys.
No point in meticulously cleaning the chain while having all that crud amassing around the pulley wheels, I reckon.
Anyway, thanks for the tip, Jer.
Always interesting to see with what somebody else's come up with.