Filling a prescription each week sounds like the easiest option, even if it means having a rest day or two each week to give the pharmacy time to stock it. You may be able to email a copy of your prescription a week before you arrive so they have time to order it in for you.
I have been travelling for 8 months with an injectable medication taken monthly for chronic migraines. I am from Australia and the medication is heavily subsided so it only costs $20/month but overseas with a prescription it costs around $900/per month as reciprocal health agreements don't cover this drug. Needless to say I have taken my own supply from home and carry it a thermal bag bought from Diabetes Australia. This bag has two ice bricks that guarantees the content remains between 2 -8 degrees for 36 hrs. When arriving at my accommodation I store the medication in the fridge and ice bricks in the freezer until moving on again. This bag is too heavy to carry on a Camino but could be sent between albergues via courier service and stored in communal fridge each night. It would just require a bit of research to make sure you booked an albergue/hotel etc with a fridge/freezer.
Next year I plan to walk the VDLP in March where I will take my last injection, ditch the thermal bag and do without subsequent doses until arriving back home the beginning of June.
Good luck with whatever you decide to do.