I dunno, I'm a programmer and know very little of astro-physics aside from Neil DeGrasse Tyson documentaries...
I think it might be that the very fact that there is an exchange of forces which is propelling "hypothetical ultra-light inter-galactically long stick", the fact that it has a tensile strength, however light might be the same thing that holds it back from going the speed of light.

Assuming that matter is made up of particles and waves. And that a wave can be a particle & vice versa. I would think that when the super-light stick hits the speed of light for a fraction of a second it would lighten because a portion of it's matter would have dissipated as light & the friction from that energy exchange might push back on the "hypothetical ultra-light inter-galactically long stick" & bring it back to just below the speed of light... I dunno, I guess I'm thinking of breaking the speed of light as more of a chemical than physical change. Like, we can get there, but once we're there we've turned into something very different, possibly permanently... So, it might be better to work around the speed of light somehow.

Thanks for the thought-experiment!