Would equipping a {whispersilk cloak} to a {psionic entity} prevent it from dealing damage to itself?
No, it does not target itself with ts ability. So the shroud part would not effect it.
RESOLVED
There are enchantments I think that prevent damage to creatures or make them indestructable. You could use those. I don't remember what they are, and lent my cards to a friend, so I can't check
I love how the only ruling on {psionic entity} is that the artist accredited on the card is wrong...lol