Not sure whether there's a more recent thread on the Pamplona-St Jean bus route, but I got an alert due to my previous involvement on this thread, so I was curious to check it out for myself! As you say, an Alsa search comes up blank. It could be that it's just not feasible to run this route at the moment, given how few pilgrims there are. It's probably also too complicated, given the border restrictions on entering France: you need to show a PCR test taken maximum 72 hours before arrival in France as well as filling out a "declaration" form. And then you might need to show another PCR test on entering Spain again from France? Not sure how far you're going to get on one 72-hour test!
This might all change by August, but if there are still border restrictions, then one hour between your flight arrival/bus departure would be cutting it a bit fine. There's normally also a 5:30pm bus, so this might be a better option? - that is, of course, IF everything's back up and running! Otherwise Artieda
is currently doing the Pamplona-Roncesvalles route, remaining within Spain so far less complicated, at least right now.