You have to doom the Titan and be credited for the kill to the very end of its life -- that includes its doomed state. If anyone else is attacking it during the doomed state and gets the final blow, it won't count towards your challenge.
So the most reliable way is continue blasting away at the Titan even after it's doomed. There's no reason to jump off if nothing is threatening you. The core takes damage faster when it's doomed so it's worth the few extra seconds to guarantee your claim. If you doom it and the pilot ejects/auto-ejects, you'll also get credit. Those are the best ones. Nice and fast.
I've also stolen a few rodeo kills off opponents that were running Survivor. They got doomed by teammates and kept on trucking until I hopped on and finished off the core.
Good luck!

Thanks man. seems like this is going to take me a little bit to complete.