Let me tell you a story. I once knew a Cardassian, a dashing, handsome young man with a promising career. And then one day, through no fault of his own, he found himself exiled and alone, with nowhere to turn. But did he give up? No. Instead he struck upon a brilliant plan. Rather than fleeing for the rest of his life, he sought shelter in the one place no one expected him to go. In a stronghold of his people's most hated enemies. There, surrounded by hostile strangers, he built a life. And there, against all odds, against the merciless logic of the Universe itself, he thrived.
This quote from Garak to Ziyal in A Call to Arms
to me implies that although Garak's exile from Cardassia wasn't his choice, it was actually his idea to live among the Bajorans, even if it wasn't ideal. I would guess that Garak requested and was granted asylum from the Bajoran government as he "sought shelter" among them.