What does the idiom ‘I don’t want to be the Babylonian messenger‘ mean?

Perhaps it's a reference to Jeremiah 51:31 in which a relay of messengers run to tell the king of Babylon that his city has been taken.

As far as the king is concerned these are not glad tidings. So your expression might be be equivalent to "I hate to be the bearer of bad news".

But if you're devout you're meant to be glad to hear Babylon has fallen, in which case your expression would mean, "I don't want to be the bearer of good news": a position most people would envy.