Invoicing [Billing] is carried out [effected] automatically on a book-entry basis, significantly facilitating expense accounting.

I'd go for invoicing or billing rather than settlement, as Abrechnung obviously refers to the invoicing / billing of the card charges.

No dictionary references (Zahn and Woywode didn't have much to offer) - my choice of terminology is based on personal experience confirmed by relevant Google hits.

See for example:
... Even though global currency transactions are executed geodesically -- directly between
bank to bank -- the transactions still settle on a *book-entry basis* over ...
www.philodox.com/conceptdefs.html

... They have entire issues there consisting of a single piece of paper in the clearinghouse
vault which are traded entirely on a *book entry basis*, that is, only ...
www.shipwright.com/rants/rant_07.html

... Book-Entry Settlement. The settlement of securities transactions is on a *book-entry
basis* with the KSD, the main transfer agent and sole central depository in ...
www.ksda.or.kr/english/stock/kse3.htm

*Expense Accounting*. Expense Accounting allows employees to enter expense
vouchers and create expense reports. Expense Vouchers may ...