There is a day pass that covers your round-trip as described for €42/3 adults, including the buses... The Rheinland-Pfalz ticket. It's valid at any hour on Saturdays, Sundays and official holidays. Buy it on the DB app or from a ticket machine at Bacharach station.
On weekdays you must start out from Bacharach after 9 am with this ticket. Random check of a Wednesday in September shows the 10:21 - 12:38 journey to be the first eligible departure, as you outlined it. On a random September Saturday, earlier choices would also be available with this ticket (8:04 or 8:50.) Since you're starting out from such a distant base town, and you'll have about 4.5 hours round-trip for this outing, doing this trip on the weekend, if you have a choice, and leaving earlier, will allow some time in your day for other activities or sightseeing.
If you have the Deutschland Ticket, there are no weekday time restrictions, and everything is covered. Leave as early as you like on any day you like.