payment to third party

It would be a very poor lawyer who entered into such a the contract on his own behalf, rather than on behalf of his client.  The fact that it was the lawyer with whom you dealt doesn't mean that he is the contracting party, any more than the salesman who sells you your car is the party with whom you had the contract, rather than the dealership he represents.  If you have a written contract, I think you will find that it is with the lawyer's client, not the lawyer himself.  If, in fact, your written contract is actually with the lawyer, and not with his client, you would have a claim against the lawyer.  If you had only a verbal contract and you reasonably understood that you were contracting with the lawyer, not the client, you may also have a case, particularly if you can show that you wouldn't have agreed to do the job if you thought the only  party on the hook was the client.