Bivvying is a such a wide ranging experience, from near-vagrancy (bus shelter, church porch, phone box ...) to near-camping. If the weather is bad for bivvying, it's going to be much the same for camping, so actually the church porch approach might be better. McNasty used railway station waiting rooms on an extended round-Britain Audax, I recall. You can spend a lot of time looking for somewhere suitable, though. Unless you're McNasty who apparently also knows the location of every habitable cave in Britain.
For Audax Hotel sleeping, a cheap bivvy-bag (more to keep the sleeping bag clean than for weather protection) and a bag liner or lightweight bag - all in a dry-bag strapped on the Carradice. Single night only, so it doesn't matter much if it all gets a bit manky. I've only done one multi-night bivvying audax (600km, Friday night start, short sleep both nights). Anything longer had the luxury of beds, under a solid roof.