Honestly, your so-called friend has a brazen streak, and I would not place any dependence upon him paying you back the money if you lent it to him, which I hope you will not.
As a pet owner myself, I would never ask a friend to pay my vet's bill for me - I would somehow or other find the money myself, even if I had to ask the vet to let me pay the bill in installments.
I do realise that the owner of the dog may be facing either having to pay for a costly operation or to have the dog put down, but that is not your affair. He could either have taken out insurance long since for the dog, or put money aside for vet's bills.
Does he not have a car he could sell? A room he could let? Time and strength to take on an extra job?
Please do not give in to this emotional blackmail. If you feel you must offer to help, don't give him money, offer to speak to the vet about an installment plan, or what the alternatives are, but make it absolutely clear the bill is his responsibility and his alone. You could offer him two square meals a week at your house while he is struggling to recover from paying the bill, but don't give him money.