Has the client explained why they want your employees to go through the background check process? Is there a legal requirement (Trid??) with regard to this?

Without more of an explanation I would refuse.  If it turned out that it had something to do with some aspect of security around Trid, then I would absolutely refuse to go along with it.  I don't have any access to borrowers' information; nor do I have anything to do with cutting checks, etc.   I provide an abstract and recording services (if required) and that's all that I do.  

I  recently declined to do any more work for a wonderful client who is now utilizing a third party vendor for assigning their work.  The 3rd party vendor needs my credit card number, so they can instantly charge me $0.75 per rundown order.  I have no information about this vendor and have no idea who in that company has access to my information, so I made the decision to walk away from the work.

 This background check thing doesn't pass the smell test for me.  I also find it interesting that the client requiring a background check has a company all lined up & ready to do those background checks.