Difficult one...even if you are right, and even if despite this you go back, they may not pay you.
The damage has already been done so far as they will never think highly of you and won't use you again (which of course will be to your advantage, but even so...) and from your end you'll never work for them again and equally will not speak highly.
You have two choices; either you dwell on it and get even more annoyed because you believe you are in the right, or you move forward. Better that you move forward i say, make a decision, stick to it, and accept what does and what doesn't happen.
So, do you cut your loses and walk away at this point, having lost the time spent cleaning the property and cost of
materials? That's one option. They can't complain then.
You can go back and invest another 1.5hours or more, in the hope that they then pay you, but they may still not be satisfied or indeed might not pay you for any number of reasons. Add to which, you say you are in the 'right', so where does going back to 'correct' it leave you?
Or, you can stick to your guns and demand payment. I presume you will have to somehow be able to enforce the tasklist they agreed and signed to prove that they are at fault in this dispute.
I know what i would do in my own business, but this isnt about me. All i will say again is make your choice and be sure to ACCEPT what then does and doesn’t happen. If you don’t accept it then you’ll be forever angry and bitter at what they ‘got away with’. If for instance you decide not to charge for the work, say to yourself (out loud) “They are not a very nice persons and I got stung. It is MY choice to walk away as I am the better person”. Do that –make your self responsible- and you are sure to feel better for it. Whatever happens, look on it as a learning curve for the next time.
Call me a cynic, but my gut instincts are that they are experts at doing this sort of thing. A well-to-do letting agent should have it’s own preferred cleaning company (amongst other services) therefore I am always wary when people like this invite me to work for them for the very first time. Best of luck, and don’t let it get you down.