Pilgrim House does NOT have showers available. The place up the street that used to have showers for rent, is now Casa Ivar - Ivar's office. There are no showers there either.
IMHO and experience, getting a room, even a bed at an albergue is the ideal option. Even if you check in, freshen up, then depart the same day, it is still a reasonable way to go.
This said, I always tell people to plan to stay at least two nights at Santiago after their Caminos. Too many pilgrims make the same mistake of assuming they can arrive in the morning, get their Compostela, attend the noon Mass, then get the bus to the airport to depart on an afternoon plane - or train from the train station. To me, at least, this is the sign - and behavior of a tourist - not a pilgrim.
Why rush? Santiago is a rich, interesting and very historic city. To believers, it is also one of the most holy places in all Christendom. Santiago de Compostela is the THIRD most visited place of Christian pilgrimage in the world. IMHO, to rush through this seminal experience is almost a blasphemy.
I am NOT judging. But, I am advocating for spending more time at Santiago to allow the effects of the Camino you have just completed to infuse your soul and benefit you - usually in more ways than you can possibly imagine.
Hope this helps,
Tom