However, she misses, and the projectile passes Bob 1 meter to his side.
The $1kg$ projectile is not a black hole and Bob is not a black hole. That means each of them is larger in extent that the black hole each would form if the parts became a black hole.
That means there is no way to miss and have the horizons merge. This isn't obvious or simple. If you had matter coming in radially from all directions there could be no black hole until they finally form the critically small surface area.
But the point is that the critical surface area is inside one and also the other, so when the two get as close as they can get, there is no way to enclose all the mass into a smaller surface area when they aren't even touching. For the in-falling matter it can become efficient to surround all the radially in-falling parts when it is about twice the critical surface area if it is made up of a bunch of thin shelled pieces.
What would Bob experience, as the velocity of the projectile Alice shoots at him (and misses) approaches the speed of light?
So there won't be a black hole formed unless one of the two was already close to forming a black hole. But there is still an effect.
Is there some speed at which, regardless of missing, Alice kills Bob anyway?
Yes. Bob alone might not radiate. The $1kg$ projectile by itself might not radiate. Either as a test particle in the curved spacetime caused by the other might be fine. But you have two masses and the spacetime between them is curved from both of them.
This will create gravitational waves. When strong enough these can disrupts the normal processes Bob uses to self regulate the interactions in Bob's body and Bob will die.
Of course at such a high speed Alice would be dead too from the same projectile. So it isn't about whether Alice doesn't care for Bob, it's about whether Alice cares to live.
What if this was happening in a more realistic setting, like the interstellar medium?
The interstellar medium will slow down the $1kg$ mass. But the exact effect depends on the shape and construction of the projectile. It could just scoop up all the material and drag it along for the ride (and slow down in the process). So increase the mass a bit and decrease the speed a bit. It could have the medium bounce off it so that the medium parts bounce off at a faster speed than the $1kg$ mass is going (and the $1kg$ mass slows down even more) or something in between.
The precursors could cause more havoc if the much smaller damage they do happens to start a process in Bob that makes Bob more vulnerable to the main attraction. Otherwise it just requires a bigger initial speed for the $1kg$ device before killing Bob.