Are bin bags that heavy?
I presume your class 2 if so then you will struggle to find a job that has zero load interaction.
With class 2 your almost guaranteed to be lumping something.
There are a few exceptions… Trunking, car transport and flat bed stuff for building sites.
Not trying be mean but you are on agency the main benefit to that is you can just try your luck with another agency with zero notice. If I was in your shoes I’d be lumping it because in my area that is job and knock with full hours pay.
Basically I’m gonna echo what everyone else has said. Your not happy because your lumping stuff, the customer isn’t happy because they got a limper with a bad back (seems to be contagious 
) and agency is happy because they get paid no matter what. Just tell them straight or ■■■■ it up.