[GNC] Billable Expenses carried over into the new year

[Andrew]

Small business owner who is definitely not a trained accountant here, so please bear with me if none of this makes sense.
Our business regularly incurs travel expenses while performing work for clients, which we invoice to the client with no markup. We reimburse the employees as they incur the expenses and I book those as Reimbursable Expenses. When the client pays their invoice, I rebate that expense account the portion of the invoice that was expenses while the reminder of the invoice is marked as income in the appropriate client's account. I was doing this so that I could differentiate between what was truly income, and what was just a cost that I was pass through to the client. 
This really seemed to work well until we closed our books at the end of the year and started 2021 fresh and we had a client pay their December 2020 invoice late. Came in on January 3rd. So now I've got a billable expense being reimbursed to us, but for the life of me I can't figure out how to handle it since we zeroed out all the expense accounts on January 1. 
Do I set up an opening balance in the Reimbursable Expenses Account? This seems like I would be double-booking the same expense in two different years.
Do I set up a Reimbursable Expenses Receivable Account and have the new invoice go there? 
Thanks,
Andrew