Pirkei Avos

As you mentioned in your question, this can happen only in Eretz Yisro'el, because in Chutz Lo'oretz in case first day of Pesach falls on Shabbos, the next Shabbos is the last day of Pesach, and therefore there are only 6 weeks between Pesach and Shovu'os like every year. For that reason this question wasn't discussed in Ashkenaz.
In Lu'ach Eretz Yisroe'el he writes that Pirkei Ovos is started in Shabbos Isru chag of Pesach. Some had a minhag to say on the Shabbos before Shovu'os the first perek of Maseches Derech Eretz (this minhag was also brought in Noheg Katzon Yosef, and he offers bnei chutz lo'oretz also to say this Perek on Acharon Shel Pesach).
Some read in that year Pirkei Ovos like Bnei Chutz Lo'oretz, and didn't say anything on Isru Chag.
The last time the first day of Pesach fell on a Shabbos was in year 5758 and the next time will be in 5772 IYH. KAYJ opened in 5760, so this issue hasn't come up yet. When we get to this question, I believe our Rov will give us a psak.

Michael