I can picture comet-caused cooling but it's going to be pretty hard to do without a big splat.
As L.Dutch says, burning a comet simply adds heat. What you need is dust. Your comet could be old, most of the volatiles gone, it's mostly dirt by now. However, any object that can deliver that much dust is too massive to burn before it hits. It's going to plunge through the atmosphere and make a big crater, complete with major splash effects.
Ok, we need more. The engine malfunction sends the comet Earthward, the world panics. The scientists know what to do--a series of standoff nuclear detonations to nudge it aside. (An Orion drive doesn't actually care where the bombs come from, only where they detonate.) Some megalomaniac doesn't listen and a missile goes out with a big impact-fused bomb, the comet is blown to bits. The comet turns out to be very fragile, this is far more successful than expected but it's not enough--many of the bits are still a threat and deflection is now hopeless. The deflection missiles are retargeted against the most threatening bits, impact detonation.
While they have no hope of stopping everything most of it is reduced to stuff small enough to burn and since the chunks have been targeted based on the threat they pose the bigger stuff that gets through falls in remote areas. Much of the dirt that comprised the comet has been dumped into the upper atmosphere--you've got your winter.