Welcome to the forum,
@Walkingbike. Just to clarify: I take it that you will bike on the Camino Francés from SJPP to Santiago and back, and that the return journey will take about 2 weeks, is that it?
I've never heard or read that the hostels and albergues on the CF would take issue with Camino pilgrims returning from Santiago who wish to stay with them. I would consider it as absurd if those who claim to uphold the century-old tradition of the pilgrimage to Santiago want to turn it into a one-way street. Those who created this tradition centuries ago did walk to Santiago and they did walk from Santiago back home. It is only today's Camino peregrin@s who are too busy to do so.
There was a thread a few weeks ago when it became known that albergues on a different Camino in the south of Spain refused to allow people to stay with them because they walked in the "wrong" direction. However that was not the main reason. The main reason was that they had not started in Santiago but in Granada far far away from Santiago and were therefore regarded as not belonging to the target group of these volunteer supported albergues as they did not meet the requirements of a "pilgrim" profile.
During your timeframe (last week of September and first week of October) things on the CF are still in full swing and you can expect albergues and hostels to be open.
Buen Camino!