Maybe rethink the gun. They have these things that use rapidly (but, crucially, not instantly) expanding liquid nitrogen to propel sensitive payloads real fast. I saw a study about using them to throw bombs on mountains for avalanche prevention. That solves the temperature problem and maybe the brittleness problem.

Also, there is an ice-adjacent material that could make the bullet named pykrete which is more durable than pure ice (Mythbusters made a boat made out of it one time). That defeats the purpose of an ice bullet somewhat, but who knows, the added sawdust might just confuse the forensics.

Though if you really wanna kill someone with an ice projectile, a big hunk of ice thrown from a sling is more than enough. Bonus points if you give it two sharp edges like the romans did.

I spent way too long thinking about this.