I've been serving as a bishop for two years, but I am not one of the tech people from the church, so I'm willing to be corrected if I am wrong. My experience has been that there are two steps to processing the reimbursements. First, two people need to approve the expense--either two members of the bishopric or a bishopric member and a clerk. The person approving the expense cannot be a payee of the expense. If the bishop was not one of the approvers, then he must also do a final review, even if he or a family member was a payee. So, in your case, the bishop should just go ahead and review the expense and click the "Reviewed" button.
I'm actually glad that I always get to do a final review because I know that I'm the one that is going to have to face the stake auditor every six months and I want to be able to see with my own eyes that the clerks and counselors filled out the expenses correctly.
A separate but related issue that sometimes comes up is whether or not the stake president needs to review an expense in which the payee is the bishop or a relative. No, the stake president only needs to review it if the bishop is the payee of fast offering assistance. If it's just a reimbursement for an expense, then it would just need to be reviewed by a bishopric counselor and a clerk (or two counselors) and then, as always, the bishop himself needs to do a final review.