There's about one particle of dust per million cubic metres. c is about 300 million metres/second. So even at 0.5c that's still a lot of particle collisions per second, each having significant kinetic energy.
Basically it would be like flying through explosive sandpaper. Each dust particle would be reduced to plasma, which creates problems of its own.
If you're accelerating there's also the Unruh Effect, which will raise the perceived temperature. By a lot.
There's no way to make this work with any kind of engineering we know about today.
The Unruh effect is theoretical, and no evidence at all has ever been found that it's real. It literally exists as nothing more than a hypothetical mathematical model, that also happens to be debated by others who know enough to effectively debate it, and disagree.
Mostly that it's plentiful, ablative, expendable, plus good radiation shielding (yeah, cosmic ray protons are really going to mess things up at relativistic speeds too unless there's enough mass to stop them).
It's one of those cases where you have very small numbers multiplied by very large ones. The actual risk is hard to intuit because there are so many orders of magnitude involved in both directions.
In any case it's probably a moot concern as long as we are living under the twin tyrannies of Newtons Third Law and the Rocket Equation. Building a rocket that can accelerate constantly and noticeably for weeks, months, or even years on end in order to accelerate up to a velocity where Relativity starts to matter requires an absurdly large rocket. Like converting the mass of Jupiter into rocket fuel to make it to the next habitable solar system in a couple of centuries level of craziness.
But it's also very big, and GP doesn't even specify how far of a trip they're asking about nor how small a meteorite.
"Extremely" and "very" don't cut it here. This is beyond the human ability to guess. You'd actually do at least some back-of-the-napkin math to give a real answer, and with a far enough trip, the answer may well become "Almost 100%".
