Hi Alex,
Both sound like great options so you can't go wrong. It just depends what you're looking for. The CF will be busy but it's a great experience in its own right. You'll meet lots of pilgrims of many different nationalities. Great fun! The CP is a way quieter route from what I've read although gaining popularity... it's more remote in places and very beautiful
I think I did the CF in about 35 days including 1 rest day in Burgos with another 5 days to walk to Finisterre. It was a great experience and I met some fantastic people but I prefer quieter caminos /hiking trails these days
I've hear people can walk from St Jean Pied de Port in 28 days so what you're thinking of doing is doable.
You'll have more than enough time to do the Primitivo and Finisterre caminos in a month. I've heard that the Camino San Salvador from Leon to Oviedo (and which takes about 5 days or so) is meant to be tough but stunning. Some people on this forum have done it so there should be some info on here. Might be worth looking into if you are an experienced and fit hiker.... ???
Whatever you decide I'm sure it'll be fantastic either way
Maybe our paths may meet along the way. If not, have a good one and buen Camino!