Since you already have a working itinerary with dates, I assume you have searched the usual booking engines and found no hotels within your budget in any of these areas. So this is a question about whether hostels are a decent choice, right?
You should have no problem finding a family room with ensuite bath in an official youth hostel (DJH association.) DJH markets heavily to families. We've found the facilities to be satisfactory in most ways for us. But they also market heavily to school groups, whose pre-teen clients tend to "take over" the hostel environment and turn it into their own private playground (IME that is.) So a hostel stay during school break periods will normally be preferrable to a stay during instruction periods (when these week-long school outings take place.)
We stayed at this hostel in Diez, near the Middle Rhine Valley, during a winter break period for 2 nights and thought it was great. Over the years, I/we have also housed ourselves at the hostels in St. Goar, Bingen, Bacharach, Bernkastel, Mainz, Cochem, Trier, and Saarburg, all in the "Rhineland-Palatinate" (the state where the MRV is located.) Here's a map of Germany with all the states. Click on the map of Rheinland-Pfalz (in the southern half on the western border) to find hostel listings that surround the Rhine or any of the other states to find that state's DJH hostels.
https://www.jugendherberge.de/en/find-a-youth-hostel/
Summer break in the schools of the Rhineland-Palatinate doesn't happen until July 24, so there'll be a fair amount of risk that you'll be booked at a hostel not just with individual travelers and families but also with the noisy school groups I've mentioned.
All that said, your best option is likely to be a Ferienwohnung, which helps you save a little on your FOOD budget as well since you can prepare simple meals and snacks "at home" whenever you choose to. Some that are bookable for 3 persons will have 2 bedrooms, others 1 bedroom + a "Schlafcouch" or Schlafsofa" (sleeper.) There are many ways to find these places commercial sites - booking.com, hrs.com, etc. - and the accommodations pages for specific towns/regions... I'll list a couple of these for the Rhine below.
Rhine Valley
Boppard