Another partial answer.
Tl;dr: kissing had a social role in Judaism that was inherited into Christianity (as osculum in the Vulgate), where it even had/acquired a ceremonial role (not sure if this one existed among Jews). Maybe—and this is pure speculation—this ceremonial role was what later made the word osculum a matter of respect and created the need for another one for its more common use. Maybe basium was a matter of pudity, but later extended its semantic field to encompass any kind of kiss.
Kisses (not those between lovers) seem to have had an important role in Jewish culture. At least it was so in the I century and was inherited into Christianity.
The Clementine Vulgate lists 53 occurrences of osculum, of which 38 are in the Old Testament, 9 in the Gospels, and 6 in other books of the New Testament. In turn, no occurrences of basium can be found.
Considering these occurrences, a word of two can be said about the role of kisses in I Century Judaism/early Christianity:
- Kissing welcome to a guest was somewhat expected (Lc 7:45)
- Kissing even the feet of someone with a higher social/religious rank was not weird at all (7:38,45)
- A disciple kissing a master was a common, yet warm, greeting (Mt 26:48-49, Mc 14:44-45, Lc 22:47-48), and was the sign chosen by Judas to betray Jesus
- A kiss was also an obvious sing of affection from a father to a grown-up son (Lc 15:20)
After the Ascension of Jesus, the kisses mentioned in the New Testament belong to two classes:
- When St. Paul paid what he knew what his last visit to the Ephesians, they were sad, and wished him farewell with kisses
- St. Paul ends four of his letters, and St. Peter one, commending their readers to greet each other in osculo sancto
This osculum sanctum became part of the Mass and was later known as osculum pacis. After centuries of social change and enculturation of Christianity in different times and places, kissing was not always the most appropriate sign of fraternal peace (perhaps sometimes too intimate) and so the signs—and name—adapted to what is today known as signum pacis (which may still be a kiss, a handshake or even a hug), and is observed on Sunday Masses and optional the rest of the days.
My intuition is that this ceremonial meaning before and during the formation of Romance languages might have triggered the need for another word for the common kiss: hence beso in Spanish, bacio in Italian, baiser/embrasser in French, beijo in Portuguese, etc.