The popular theory of "acceptable trianitian baptism" is not the mono-voice in Orthodoxy. In fact ,I think the question should be "If exist real mystery out side of real church"? Can mysteries be kept only by "right mothod of administration"without ture faith——Orthodoxy? If so,then the mystery is some magic can "live along" outside of the life-giving body of Christ——His church?
It's will be more gracious for your family to be baptized together by the orthodox way(triple immersion). If the priest or bishop reject,just try from otherone——bishops are not gurus,and the obedience towards them are not samaya vow.
FWIW, I'll repeat the Orthodox doctrine on heterodox baptism so that it's clear for this thread. There is no baptism outside the Orthodox Church.
However, upon one's reception into the Church,
we believe that an Orthodox chrismation is capable of filling the empty shell of the heterodox baptism if the baptism was performed according to proper form (i.e., triple immersion in the name of the Father, Son, and Holy Spirit). This oikonomia
of recognizing chrismation as sufficient for reception of those baptized outside the Church (to be seen as the exception and not the rule) renders the original baptism Orthodox, making (re)baptism unnecessary.
As to the idea of "this bishop won't baptize me in the Orthodox way, so I'll look for a bishop who will" ... A very dangerous attitude, indeed, since you, a potential convert, are setting yourself up to know more of what you need pastorally than do Orthodox bishops. Is this not the epitome of the prideful convert?