Meanwhile the utility of the mugger's finger is questionable. The pain of losing the finger is the only real cost. If they are just a petty criminal, the loss of their finger will probably reduce their ability to commit crimes and prevent him from inflicting as much suffering on others as he otherwise would have. Maybe losing his finger actually increases utility.
Bentham: "I'm sorry Mr. Mugger but I am on my way to spend this 10 pounds on a supply of fever medication for the orphanage and I am afraid that if I don't procure the medicine, several children will die or suffer fever madness. So when faced with calculating the utility of this situation I must weigh your finger against the lives of these children. Good day. And if the experience of cutting your finger off makes you question your own deontological beliefs, feel free to call upon me for some tutoring on the philosophy of Act Utilitarianism."
Any other scenario and Bentham clearly isn't a true Act Utilitarian and would just tell the Mugger to shove his finger up his ass for all Bentham cares. Either strictly apply the rules or don't apply them at all.