Cool, go for it. It’s not as much hassle as people might tell you.
We use washable nappies in general life and our experience of using them on the camino was a little problematic, but not terrible. The plan was to take a limited number of nappies, hand-wash them as we went and dry them as much as we could on radiators at night, and off the back of the rucksack during the day. It kind of worked for a while but because we went in winter, there wasn’t much sun-drying during the day. We were always short a few clean dry nappies. In the end we parked the washables in a pension and used disposables. We felt bad about that but if you go in the warmer weather, you should have no problem rotating clean, dirty, wet, and dry ones.
As regards sleeping, most of the pilgrim hostels wouldn’t take us with a baby, because it is all dormitory-style communal bedrooms. That’s understandable. We generally had to stay in private pensions (cheap, family-run small hotels, basically). It costs more, of course, but it also meant we had privacy and the baby wasn’t troubling others.
It’s great fun with a baby, everyone will love you. We are going back to do some more later this year, possible by bike the next time. Let me know if you need to know anything else.
Jim.