I walked Route Valcarlos at the very beginning of April when it wasn't clear that Route Napoleon was safe. Some took Route Napoleon, and many had a long day of it, with some arriving at Roncesvalles after 8.00pm. But it was clearly possible even when they did the last couple of hours in the dark.
Next time I do the CF, I plan to take two days rather than carry a tent and camp out. Why? If you are thinking of taking a tent for this or any other stretch, you would need to contemplate having food, cooking and eating utensils, water for the evening, etc. Very quickly, you are carrying much more than just the weight of the tent. Its not impossible - I have walked with over 20kg on my back, but its not necessary on the CF or any of the other pilgrim routes so far as I can tell.
When I walked, only one private albergue didn't have space when I arrived. Everywhere else, there was always room available, if not at the first alburgue, one a bit further into town. Yes, there are snorers, but I don't think lack of beds will be a problem at that time of year. My own view is that carrying a tent is a big effort just to avoid snorers when a very much lighter set of earplugs might suffice.
As for staying an extra day, it is possible to plan a short walking day before and after most of the major towns along the CF, which would have much the same effect. The only place that I recall where you couldn't stay if you had just walked from the previous town was at the parish albergue in Granon.
My last point, and that is whether you need to carry a 15degF/-10degC bag. Certainly if you were planning to camp at this time of the year, it could be right. But I don't think you would need that level of insulation if you were staying in albergues. I carried a 5degC bag in April, and it was enough for the places that weren't heated (at Zubiri, which didn't provide blankets in the old gym, the parish albergue in Granon and at Espinosa).
