Randy's right. You can't "raise the bar" above what Reg E says. Another way of saying it is, "you can't take people's rights away from them". Reg E says they don't have to sign anything. Your processor can't change that rule. If your processor won't investigate without a written statement, then you still have to live by Reg E. So you're stuck with investigating it yourself or giving the customer their money without investigating it.

Taking this one step further, if you don't investigate the claim due to a customer not returning a signed statement, you could be criticism by an examiner for a UDAAP violation. As David pointed out, this practice is "raising the bar" above what Reg E says, which is substantially similar to what the Fed and FDIC said when they cited banks for UDAAP violations for requiring police reports as a condition of an investigation. Their reasoning was that the requirement of a police report was both beyond what Regulation E requires, but also above and beyond what was disclosed in the error resolution notice. Therefore, requiring customers to jump through more hoops than what was disclosed (and what Reg E requires) can easily drift into the wonderful world of UDAAP.