Is it possible to make a JE line billable to a customer?

Not possible.

@sam1128 thank you. I didn't think so

If you have an AR account on the debit side (credit side is your revenue or deferred revenue account for example) with the customer name in the Name field, that JE line is essentially an invoice and you can apply customer payments against it.

Yes but it's not native record for it. You should use invoice instead also journal don't have paid, open and due dates so not part of ageing.