Hi Alq. As far as the St. Jean de Luz partido season, every year it is from about July 1st to August 30. Keep an eye on the St. Jean de Luz site. They usually announce the exact dates fairly early. I'd start looking around February and bookmark the site. Biarritz runs Gan't d'Or at about the same times. St. Jean de Luz usually runs its programs on Tuesdays and Thursdays, Biarritz on Mondays and Wednesdays, so there is always Jai Alai going on on the French side of the Pyrenees. They usually have other tournaments also, particularly at Hossegor. Google Hossegor Jai Alai or try Hossegor Cesta Punta on Facebook.
The Spanish side of the Pyrenees is a litte more problematic. Jai Alai World Tour doesn't seem to me as organized as Cesta Punta Pro Tour. They come out with schedules later and this year had sponsorship problems. But there is almost always a multi-week Grand Slam at Gernika. These matches are usually on Saturday nights and are accompanied by concerts. The actual scheduling wasn't announced until very late this year. Hopefully next year will be better. Hondarribia usually also has a multi-week Grand Slam Tournament that is held on Fridays. And Zumaia (home town of Goiko) had a multi-week tournament this year, but very late, perhaps out of your time frame.
To summarize, if you are planning to travel, it is probably easier to coordinate with St. Jean de Luz and Biarritz as they tend to announce earlier. But the place I'd want to check out, if I could, would be Gernika. Follow the Jai Alai News Facebook Page, it is a very good source of general information about Cesta Punta in the Basque Country. Most of its articles are in Castilian (Spanish) but it's pretty easy to set any browser to auto-translate, if you aren't fluent in the language.