I would definitely recommend at least a day in Santiago, if only to rest up before the very long train ride (probably nearly as long as your flight from Paris!). There is a daily train from Santiago to Hendaye, which takes about 11 hours. You get to Hendaye after 9pm and there are then no onward trains until the following morning, so you will either need to stay the night there, or get off the train earlier (eg at San Sebastian), stay there and then continue on to Biarritz the next day (changing at Irun or Hendaye). Either way it's not possible to get from Santiago to Biarritz by train within a single day. It's possible there are late buses which you could take onwards from Hendaye, but you would need to stay somewhere, either there or Biarritz. If you have more time, personally I would break that 11-hour trip somewhere - we stopped overnight at Leon and felt better for it, but then we still had to stop again in Hendaye. Alternatively I think you can take a night train from Santiago that is heading towards Madrid and then change somewhere (in the middle of the night), onto a train heading for Hendaye the next morning. A bit more complicated! The best thing is probably to wait until you are there, or nearly finished your camino, and see what you feel like doing.
Good luck with the planning and buen camino!