I think it's largely a matter of how much advance warning we have. If we discover an object hundreds of kilometers in diameter on an orbit that will intersect with Earth in 100 years, then we'll have a better opportunity to nudge it early on so it misses entirely or develop a big enough bomb to turn it into a pile of gravel and ice chips that will (mostly) miss the Earth.
The biggest fusion bomb detonated on Earth was Tsar Bomba at 50 megatons, though the full yield is thought to have been more like 100 if the Russians hadn't deliberately nerfed it over radiation concerns.
Is there a theoretical upper limit to the yield of fusion bombs? I assume no one is building them bigger simply because there's no realistic military use for such things that wouldn't be better served by smaller accurately targeted nuclear bombs. Tsar Bomba was detonated in 1961, and no one has seen fit to repeat the experiment, though I suppose one or more of the major nuclear powers may have a modern warhead with equivalent yield that they just haven't tested or announced to the world.
Wikipedia is saying that there's thought to be a practical limit of around 6 megatons of yield per metric ton of bomb mass [1], and actual nuclear devices have achieved a little over 5, so I guess after some point there's not much reason to make a bigger bomb when you can just make two smaller ones.
This scenario is starting to sound a bit like a long-running Factorio game where the goal is to launch a rocket once a minute. With, say, a one year lead time could the economies of Earth launch a starship-style rocket with a half-dozen or so Tsar Bomba sized warhead once per day indefinitely? I think so. Would it be enough to destroy a dwarf planet? Probably not, but maybe it could knock loose enough chunks to nudge it into a slightly different orbit.
[1] https://en.wikipedia.org/wiki/Nuclear_weapon_yield