I would have to consult my library for a more intelligent reply, but from the top of my head, I can tell you (if memory serves right):
1) I have never encountered any Proper Mass or Office of the Holy Infant Jesus. The Flight into Egypt and the Finding in the Temple had their proper Masses and Offices at one time (and perhaps they still do in some localities).
2) If I am not mistaken, the Feast of the Holy Name was traditionally regarded as the Feast of the Miraculous Infant Jesus of Prague, and the Franciscans, who are great propagators of this devotion, do not appear to have a Proper Mass for the Holy Infant, though they do have a proper Sequence for the Feast of the Holy Name.
3) I am unaware of any proper Masses or Offices in Latin American and the Philippines, where the Holy Infant Jesus was honored under various titles in numerous dioceses and localities.
4) If you would have a Mass said in honor of the Holy Infant at a traditional Chapel, it would be the Votive Mass of the Holy Name, since the Masses of Christmas and Epiphany cannot be said as Votive Masses.
I will have to research for further information.