Does the Pup count toward the final dice pool for Final Salvo?

Not clear by RAW. If it's part of the last ships standing I'd say yes. If it's glorious mutual destruction I'd say no.

It would only add its dice if it is in play at the time the game ends in a tie situation where both sides have currently killed an equal number of points.

In the MAD situation it isn't part of the squadrons so it wouldn't count. Now if it still happens to be standing when everything else is destroyed then there is no need for final Salvo as the Pup standing alone wins.