I see the unit now, your easiest mod is probably a thick thermal pad to transfer heat to the chassis. It might work out well if you combine that with those graphene heatsink stickers. (SSD > graphene heatsink sticker > thermal pad > chassis)
I have a feeling you might not have a good time with a standard SSD heatsink/dissipator. Those usually work best with at least a little bit of airflow. It might not be much different once the heatsink fully saturates. The heat needs to go somewhere