I've dabbled in FTE over the years but my career path is very out of date. So just one comment about a young engineer I know who started off somewhat like yourself and who is now working as a full-time FTE overseas, probably about 10 years after graduation. Persistence and with that goal always in mind gave progressive small but steady steps in a career progression towards it. Plus the course at ITPS in Canada - there are others.
I am sure that others will pop up with advice for some-one with a non-military background if they can be enticed away from the flight test forum.
Avalon at the end of February is a good opportunity to connect to people about this.