Allow me to offer this perspective. June 29 is the Feast of Sts Peter and Paul and is a day of obligation for Byzantine Catholics, but is not for Romans. Would one argue that our obligation would not be met if we attended a Roman Catholic Mass that day?
How could one be more righteous if he attended neither?
But I would say the same thing here as well. The understanding of the role of Peter in the EC Churches would be different from that in the RC Church.
The emphasis here on "obligation" is already a Latin import and non-starter to begin with. Eastern Catholics are called upon to attend the Divine Liturgy to celebrate ALL the major feasts of the Lord and of the Mother of God and the Forerunner throughout the year. And if our bishops have established "days of obligation" in imitation of the West, then EC's should just return to the original tradition of making every effort to attend the Divine Liturgy for all those feastdays (I also understand that the RC Church in the U.S. has basically moved the majority of the important feasts to Sundays - that is really breaking with Tradition for heaven knows what reason!).
In addition, an EC parish which is true to its heritage (as Vatican II has asked us all to be) would have a different take on the idea of primacy more in keeping with Eastern Christian collegiality and the role of the Patriarch/Primate etc. That would/should be reflected in the sermon. I would imagine, as has happened before, that Roman Catholics attending an EC Liturgy and hearing about our theology of primacy would be somewhat shocked and dismayed . . .
It is best to keep to our Particular spiritualities by attending Divine Liturgy in our respective traditions as far as possible.