7 Passenger van is about $1800. Do you realize that according to ViaMichelin for a Sedan, fuel alone, not including parking, for that trip is $500. That was for a sedan; does a van get as good mileage as a sedan?
What is the relationship of the six people, six unrelated adults or 2 parents and 4 children, what ages? I'll assume 6 adults.
BTW, FRA-Rothenburg, Rothenburg-Prague, Prague-Berlin, Berlin-Bacharach, Bacharach-Cochem, Cochem-Frankfurt are six days. Why are you looking at a five day rail pass? Or was that a typo?
Day 1: Frankfurt to Rothenburg, one RMV Gruppentageskarte (€23,60)+ one Einzelfahrschein (7,15) gets you to Kahl am Main. One Bayern-Ticket (€28) plus one Bayern-Ticket-Single (€20) gets you from Kahl to Rothenburg. That's €78,75, $106.31, for the first day.
Day 2: Rothenburg to Praque, Bayern-Tickets (again one 5 person Bayern-Ticket, €28, and one single, €20 gets you to Furth im Wald, on the German-Czech border, 6 times €11.69 for Czech Rail tickets to Prague. €118,16, $159.52.
Day 3: Praque to Berlin, 6 Europa-Spezial-Tschechien Tickets at €29 each, €174, $234.90. Build in a stop in Dresden.
Day 4: Berlin to Bacharach, 6 Sparpreis tickets at €138, $186.30.
Day 5: Bacharach to Cochem, Rheinland-Pfalz-Ticket, €28, plus RL-P-Single, €20, €48, $64.80.
Day 6: Cochem to FRA, Rheinland-Pfalz & RL_P-Single plus 6 local RMV tickets at €3,80, €70,80, $95.58.
Rail tickets for all SIX day come to $847.41, compared to $2160 for Railpasses and $1800+$500 = $2300 for the van (plus parking).
BTW, I assume the $1900 was the group discount (6 or more @ 50% off of full price). That's not usually the best deal.