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No you won’t.
C’mon they have a video showing it! It has to be real!!!
Ship gets lighter as more fuel is destroyed. Ship gets heavier as it gets closer to light speed.
At 90% of light speed, the ship’s mass would be around 2.3x its rest mass.
I haven’t looked at your calculations in detail, but you seem to be missing that important point!
the mass increase due to relativistic effects is really really small, significantly smaller than the weight loss due to fuel consumption, which btw is described by the Rocket Equation.