The reason Godzilla didn't send the other Titans after Mechagodzilla was sort of explained in Godzilla Dominion. Godzilla ordered the other Titans to go into hibernation because they were constantly getting into conflict with each other and humanity and potentially posed a threat to the world. Plus, due to the fact that Mechagodzilla was being controlled by Ghidorah's brain (in a way being a reincarnation of him) Godzilla probably viewed it as a continuation of his personal rivalry with Ghidorah.
As far as Godzilla not sending them after Kong, again it wasn't their fight. Godzilla's rivalry with Kong was a remnant of the ancient war between their two species, presumably the other titans had no part in that rivalry.
"Neither beast nor man. Something monstrous."