Hola and yes you are posing a difficult question. Now about the ALSA buses, who told you that they will not take your bike?? When I last researched this question the ALSA advice was that up to two bikes could be carried on each bus. You have to pay an additional fee (1-5 euros I think). But that only gets you to Roncesvalles - as I do not recall ALSA going over the border into France. Do a bit of research and see what comes up. You have the time.
To those who offered advice about Spanish trains - RENFE here I can speak from experience. If the bike backs down to 1200mm or less in length and about 250mm in width it can travel on the media distance trains. Very few AVE trains have large freight areas. If your bike is fully assembled then you need the local trains. I had to wait for 4 hours in Sahagun in 2015 to get to Leon (it was raining very heavy, 80 km cross winds and the temperature was below 10c. Not good bike weather).
@Eben Jacobs - when you do eventually get to Santiago there is a great cycle shop (Veloceopedo - I think at that's the spelling) and its right on the Camino trail was you enter Santiago from the Monte del Gozo (about1-2 km past the Autobus station). I used them back in 2015, they told me where I could get the bike cleaned, they packed it up and arranged transport (back to Madrid in my case), it had to go to a hotel - not directly to the airport.
Hope this helps all concerned.