There are occasionally quad rooms, but they are not common. Your best bet might be a Ferienwohnung (vacation apartment), which will often have one or two bedrooms plus a sofa bed in the living room, but they often rent for longer periods of time (around a week).
As for track noise at any place right on the tracks in Bacharach, like Kranenturm, some people, like Steve, say the noise didn't bother them, others can't stand it. Realistically, however, the worst noise is when a regional train from St. Goar stops at the Bacharach station with brakes screeching, or a train comes from Bacharach station to St. Goar with the motors laboring to get up speed. And those regional trains don't stop from about 23:30 to 5:00. Other than that, night time traffic is mostly freight trains and a few night trains that "whoosh" right by with less noise than one that stops at the station.
In Bacharach, there is a nice place, Pension Im Malerwinkel. It's closer to the back side of town, away from track noise, in a very pretty setting. The two apartments they show as being for 2 people and renting, it appears, for a minimum of 4 nights. The also show a Ferienwohnung but don't say how many people it accommodates. You might write to them and ask if there is also room for a temporary bed in the triple (Dreibettzimmer).
As for Kent's comment about "cheap but nice", I guess it depends on your definition of "nice". If you demand 24 hr room service and towel warmers in the bath room, cheap might be a problem, but Germans, at least, seem to have a very high standard. I've never stayed in a room that wasn't clean and well maintained, no matter how inexpensive. Small maybe, but never a dump. And by the way, instead of cheap (bellig, in German), I prefer preiswertig (economical).