Should we go after the vet for wrong diagnosis?

Our 7 year old cat recently fell ill. He was coughing, lethargic, not eating. We take him to a vet who promptly, and with out an xray or chest examination, diagnoses him with allergies. He was given a steroid shot and the visit cost 89. Our cat steadily deteriorated the day after and we called the vet who told us to wait until morning. Tonight, he was panting, gasping for air, and coughing up mucus. We took him to an emergency (and different) vet who told us unfortunately the cat could not live for more than a few hours. We made the painful choice to have him euthanized. We were told he either had pneumonia or FIP (a parasite). We were told the steroid shot may have made it worse. This all cost 152. We now are wanting to get our money back from the first vet and maybe the cost of seeing the other vet and the euthanasia. The loss of our pet is super painful and we feel it came all too quickly. Should we demand the vet pay us back for our expenses? If so, how should we go about this?