There is a consideration if this is a bilateral contract. The promise to perform the search in return for the promise to pay for it constitutes  consideration.

If it were deemed a unilateral contract, the abstractor's performance of the search is both his acceptance of the offer and the consideration for the contract on his side.

The issue of non payment is a normal ploy in a contract case in which there is an alleged problem with the quality of service. If the client had paid for the search it could be deemed an admission that there was no problem with the search. Backing off on the demand for the fee by the abstractor could be deemed an admission that there was a problem with his work. 

I do a lot of contract work, and the client's continued acceptance of services after the date of the alleged service deficiency kills him in court, provided he continued to accept the services with knowledge of the alleged service deficiency.  If however he continued to accept the services, but was not made aware of the alleged service deficiency until a later date he has a defense/rebuttal to the continued acceptance of services.

In so far as the issue of standing is concerned the client must be an aggrieved party in order to have standing to sue the abstractor. In Connecticut the plaintiff can not plead someone else's damages. If however, the client is on the hook to his client, he then has damages, is aggrieved and has standing to sue. If the client is sued by his client, he may join the abstractor to the suit as a co-defendant or third party defendant and assert a claim against him.
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