Actually, my dear Swan (or "Lebed" in Ukrainian . . .), All Eastern Catholics believe they are fully Orthodox and we even use the term in the liturgy!
The small group you refer to are those who may wish to understand Rome's authority in terms of the first millenium of Christianity - and that is their perspective (and/or issue?).
Orthodoxy was in full communion with Rome for the first 1,000 years (and more?).
Eastern Catholics see themselves as "Orthodox" in terms of the faith they hold and "Catholic" in terms of the Church they belong to. There are differences and, more than these, the fact of the break between Catholicism and Orthodoxy.
But we see ourselves as having the fullness of the original faith and Church - agree with us or not.
And "Byzantine" is actually a later term. The Byzantines of history wouldn't have called themselves that. They simply called themselves "Romans."