Hola Julie
Walking over the Pyrenees, either by the valley or the peak, is in my opinion a challenge for any pilgrim starting in Saint Jean. It is around a 1450 m or 1050 m accent over 22 km. In other words, it is uphill
My experience is that most people can do this. Either in one or two days, if the weather allows it.
It's a magnificent and demanding experience.
What makes it hard is not the accent, but that it is the first day, when departing from Saint Jean.
Even with preparations and training, the body most likely will scream; what are you doing to me?
Once over, and in Roncesvalles, it will become much easier for several days.
Time to allow your body to cope with your plan.
One way to think of the Camino is in weeks instead of 8-900 km walking.
First week; hard physical experience, friendships build, inner change.
Second week; joy of overcoming the first week, landscapes and nature changing, food, comfort accepting the unexpected.
Third; Injuries, see you laters and hellos, impatiences, strength, injuries.
Fourth; Amazement, determination, 40+ km, fatigue.
Maybe other members can add to the list.
In other words: As you are experienced in walking, I feel confident enough to say again; get yourself to Saint Jean and take it one day at the time.
Ask the locals for advice about the Camino, good restaurants, history, weather.
Be wise, and keep company when you feel you need it. (Do not pass the Pyrenees alone. Why would one?
Sharing is dividing the burden and doubling the experience).
About your air ticket.
It sounds to me as if you have options to go either 4 or 5 weeks.
To book a long flight ticket late can be very expensive so this should obviously be avoided.
If you fly internal Europe, late tickets will not vary much whether you book early or late (buy insurance to reschedule cheap).
Anyhow. If you have the option, then book now; give yourself 5 weeks.
4 weeks to Santiago, couple of rest days, walk to Finisterre/Muxia.
Allows you to take it easy along the path.
Or just go with your gut feeling; 4 weeks.
I walked it comfortable in 27 days and had an experience of a lifetime, so yes, you are on the right track.
Your Camino has already begun. Take it one day at the time.
Buen camino
Lettingo