My buddy swears he can do it on PC, but I can't seem to figure out how on my Xbox One. Can I without the perk and if so, how?
I'll convert this from a comment to an answer, so it's helpfull to anyone regardless of platform.
It's worth noting on PC, you can hold down X then tap E to eject manually, but it will destroy your Titan even from full health.
On Xbox One, hold down [d-pad up] and repeatedly tap [X]
For Xbox 360, hold down [d-pad up] and repeatedly tap [X]
You should see an icon of the ejector seat on the bottom right of the Titan hud. Hold this button and repeatedly tap your 'interact' button.
See this screenshot for the HUD item in the bottom right.
Several of my friends have had trouble ejecting during the 'doomed' period. It turns out they were holding 'X' (for Xbox One), instead of tapping it.
From the title, I thought this might apply, but the text says otherwise. I'd thought I'd say this anyways, in case anyone else has a similar problem.
Regular get out - hold 'X'
Eject - tap 'X' when prompt appears.