I don't think it is any sort of bacterial/viral illness as she seems otherwise quite well. It's just before laying the egg she starts to slow down and put her tail down.
Unfortunately that's what most people think. Yes, they usually DO seem 'quite well'.

They're masters of pretense and can even seem quite healthy looking when they're dead, lol.
Just because a symptom doesn't make them obviously look like they're dying doesn't actually mean it's not taking them towards that outcome, and doesn't mean it's not serious; unfortunately too many people only pay attention to the symptoms of dying, having dismissed all the previous symptoms of disease, and that's always what they say, in some variation or another --- "I don't believe it's a disease because they seem well". Later on they tend to have tales of chooks that 'just dropped dead without warning'. But the warnings were almost always there, just so misleadingly mild in appearance that people ignored them.
Quite often the only symptoms you have are small ones (like soft-shelled egg laying, that's one symptom of some serious diseases that occurs sometimes in absence of all other symptoms, initially).
Most people dismiss these seemingly minor symptoms of something being wrong, because they expect to see the animal in the dying stages and in great distress before they can confidently make a diagnosis of something being wrong. By the time you've let it go that far it's often beyond fixing. Abnormal events should always be taken as being potentially serious, after all something has clearly gone wrong to produce these symptoms. Many things can go wrong and share the same symptom, both mild problems and serious ones alike.
This is not the sort of occasional soft-shell egg laying you can dismiss as being non serious, if you can't figure out what's wrong with this hen you're fairly likely to lose her, and whatever is causing it is potentially contagious. Hopefully not but always best to be aware of the possibility. Some serious viral diseases, among other types, have this as a symptom.
This is not a mild pattern of symptoms either. One soft shelled egg every few months is still worth noting and addressing; a regular pattern like she has, and the deformed egg membrane as shown, is serious.
Sorry to flog the point, but plenty of people on this site have lost chooks by following the exact same thought pattern you described in your post.
Best wishes with finding out what's going on and fixing it, hopefully without loss of this hen or others. For what it's worth, I'd treat her as though she had a virus, just in case.