Hola
In my experience you will not need a tent.
A few years ago I walked the Valcarlos route from Saint Jean to Ronchesvalles in early February, as the high pass was closed. The locals told us not to walk the Camino trail but go by the asphalt road, due to bad weather and snow.
The asphalt road is by the way much longer than the trail.
It was quiet a hard day for our group of first time pilgrims, but we never felt at risk although we arrived late in darkness at the Albergue. (We did have lots of challenges and a blizzard is not particular fun).
If there had been a need of 'rescue' ourselves, we would just have stopped one of the many cars on the road and asked them for assistance. A tent would not have made much sense to bring, a there are many other possibilities for aid. The
Camino Frances is not a walk into wilderness.
That being said, I think the best advice would be to; bring food, fruits and water to last a day. Stay with fellow pilgrims and talk about your current situation to make sure all are well and comfortable, follow the advice from the locals, take the breaks your body will tell you it needs.
I hope this advice does not seem reckless, but my experience from looking at other pilgrims is, that once we are on the Camino, we exceed what we though was possible.
And the first thing anyone should pack into the backpack is a large bag of good sense.
Buen Camino
Lettinggo