| Garbage-Tier Waifu |
Yes, they can turn into a tattoo. They are no different from a normal familiar and gain the benefit of the feature.
The curse, though, would be up to DM fiat, but since the sprite does not count as a separate creature, it either doesn't have to worry about the curse, or so long as it is tattoo'd, you suffer from the curse. Hopefully your DM doesn't rule for the latter.