Welcome to the Forum.
Of course this is doable. In fact I am willing to bet you can walk past Pamplona, at least to Logrono if not Burgos if you are of the faster walking group. I am not, I walk like a geisha, so 10 days would only get me to Santo Domingo.
Is it the best route to take? It is the one in my heart. My first Camino. Mine was SJPP to Burgos. And when I went back the third time, I went back to Pamplona and walked to Leon.
Some will tell you you don't have to syart in SJPP, but for me it's a must. It is beautiful, and a challenge. It's not a walk you will casually repeate here and there. For me, it's a marker in my life.
Some will tell you to book ahead and only walk 8km the first day as they are steep 8km. But only a handful do that, since there are only so many beds, but also because, at least in good weather, and taking your time, it is quite doable.
Logrono is Spain's wine capital and has excellent public transit (bus, train and airport). Where is your flight back to the US from? Paris? Marseille? Try the rome2rio.com site, it will show you different alternatives. A flight back home out of Barcelona could work. Same goes for Burgos.
For getting to SJPP, try Rome2rio.com, but with the correct spelling for Le Lavandou.
For planning your etapas, I like the Ersoki site. Others like Gronze. You will see distances between albergues and suggested etapas. But as there are now soooo many albergues, it is very easy to make up your own etapas, making them shorter or longer.
Most important is to pack as light as possible, make sure your footware is broken into and comfortable, bring quick drying clothes and a fist aid kit for blisters, just in case. A bit of sun block and a liner as a sleeping bag.
Voila!