Looking just now at a simple line map showing A Corunna and Ferrol, and having once taken the bus from Santiago to Ferrol, I wonder if you could take a bus and get off in Hospital de Bruma, leave your big luggage in the accommodation you booked for the first night from A Corunna, and then get the next bus on to A Corunna and just have essentials for the night before starting to walk from there to Bruma.
I have just checked, as I imagine you have, and Paq Mochila does not offer pickup from A Corunna, only from Ferrol. You can easily book your daily transfer of bags with them from Bruma, and paying for a bundle of days at a time is - or was in my case - cheaper than day by day. (Although I also had my Bray to Dublin cert, I began in Ferrol as my companion needed the longer distance to get her Compostela.) Another option is get to A Corunna, take a taxi to Bruma, leave your stuff, taxi back... but I have no idea of cost, just trying to give you ideas that you probably have thought of yourself!