I don't understand why some people think the PM should pay.
If a REA finds a sale and the deal goes unco and then the buyer runs away the agent is still entitled to their commission and is entitled to another commission if they find another purchaser. As you employed them to find you a buyer which they did.
Same with the PM, you hired them to manage the property (notices, inspections etc etc) and also find a tenant. Which they did.
Yes it might of only taken them 1 day to find you another acceptable tenant, take this as a bonus and maybe even pay them a bonus for doing a great job in securing a tenant so quickly!
What if it took them 3 months would you then pay them $10,200 (850 x 12 weeks)? No you wouldn't cause you would of lost $10,200 in rent + letting fee of $10,200 or a total of $20,400.
Maybe you should be more grateful with what you have, sorry you really aggravated me with trying to hurt someone's income for doing a GREAT job when it could of been a lot worse.