David,

Unfortunately, it sounds to me as if the error lies in Quality Control within your office.  Any potential discrepancy should have been noted and the client contacted immediately.

What makes this even more alarming to me is:

1.  Why on earth would you travel 110 miles one way to perform a search;

2.  Knowing the distance and the sporadic relationship you have with this "client", why on earth would you not have verified the terms of the request and even the amount to be invoiced prior to putting the keys in the ignition of your car?

Again, both issues would fall under our Quality Control Department and if I were in your shoes, I would instruct my Accounts Payable Department to cancel the invoice and provide the client with a letter of apology for the invoicing error.  It was our QC Department's fault.  No two ways about it. 
At that point, each order placed, as it is in my company, would be reviewed for potential county error; name spellings; address spellings and any other oddity within.  There are too many opportunities to waste the time of an abstractor or to be notified of a "no find" when a little QC could have alleviated the breakdown. 

PS - whenever these errors occur and they are deemed our fault, we pay the abstractor full price.  That said, our errors are probably 1 out of 250 because of our QC Department.

Sorry but this is only my professional opinion based upon the facts you outlined.

Best of luck....JT