Wording on Dynamite:
"The fuse isn't lit... yet.
Instead of attacking, you can throw the Dynamite through a door into an adjacent room. Each explorer and monster with Might and Speed traits in that room must attempt a Speed roll:
5+ Take no damage.
0-4: Take 4 points of physical damage. (or suffer the normal effects of being defeated with a Might attack)
Discard this item after you use it."
Wording on Dog
"Take a small monster token to represent the Dog. Put it in your room. (Use a token of a different color from other monsters, if any.)
Once during your turn, the Dog can move to any explored room up to 6 spaces away, using doors and stairs, and then return. It can pick up, carry, and/or drop 1 item before it returns.
The Dog isn't slowed by opponents. It can't use one-way passages or rooms that require a roll. It can't carry items that slow movement."
The Dog cannot attack, therefore cannot use the Dynamite. You cannot 'attack' then send the Dog on a suicide mission. The Dog would simply drop the stick of dynamite at the feet of the monster or traitor, unlit and able to be picked up by the traitor if they wished to.
If you choose to do so, that is your gaming groups prerogative, but isn't in the rules.