It will be far simpler for you if you purchase a railpass. But if you are willing to purchase tickets instead and use slower train options in some places, you might save nearly $300 off your railpass cost of €628 (my low-end estimate for two is €313 or $332.) I cannot be sure about prices at this point because I do not have the information needed; travel dates and approximate departure or arrival times, which you did not provide, are crucial for accurate prices. Check DB on your own for routes, travel times, changes of train, and prices.
The following prices are for 2 adults. The lowest price "saver fares" noted below may or may not be available now for your dates since they've been on sale for some time.
The good news is that the daypass options ARE available on the spot on any day. They restrict travel to regional trains - so your trip may be a little, or sometimes a lot, longer. You must specify "only local transport" under "means of transport" if you want to find journeys for the day passes. Use them anytime on Sat or Sun, after 9:00 weekdays:
Rheinland-Pfalz ticket
Quer durchs Land ticket (Wherever QDL is suggested, if it's Sat or Sun, you can get the cheaper Happy Weekend ticket instead for travel at any hour.)
Bayern Ticket
Hessenticket
The VGN Tagesticket Plus has no hours restrictions.
FRA - Cologne (€52), Quer durchs Land day pass Buy at FRA from a ticket machine. Use regional trains. (OR pay full fare of around €100 or more.) (This is the slow, scenic Rhine River route Lee discussed. A cheaper saver fare on the fast train that bypasses the Rhine is possible - maybe for your date and hour - but a flight delay or cancellation means you risk forfeiture.)
Cologne - Bacharach (€29) Pre-purchased Saver Fare if available. (OR use regional trains only, €43 w/ the VRS ticket and the Rheinland-Pfalz ticket bought from ticket machines in Cologne, as Lee suggests.
Bacharach - Baden-Baden (€29) Saver Fare for fast trains if available. (OR use regional trains only. €52 w/ the QDL ticket from a ticket machine.)
Baden-Baden - Munich (€49) Saver Fare (OR €52 regional trains w/ QDL from a ticket machine.)
Munich - Rothenburg (€28) Use regional trains w/ Bayern Ticket from a ticket machine (or pre-purchased saver fare for €29)
Rothenburg - Bamberg (€18) Use regional trains w/ VGN Tagesticket Plus from a ticket machine.
Bamberg - Philippsthal (€52) Use regional trains w/ QDL Ticket from a ticket machine. (Plus bus fare at end.)
Philippsthal - Kassel (€27) (two normal adult fares @ €13.50 from the bus driver, local NVV fare. Switch from bus to train.)
Kassel - Frankfurt (€29) pre-purchased saver fare for fast trains if available (or for €33 use regional trains only w/ a Hessenticket day pass from a ticket machine.
Reservations or not, I suggest dropping Baden-Baden if it's not a genealogy stop and adding nights elsewhere closer to your route. There are so many nice places you might visit.