I would say you need significantly less material than most of y'all are thinking to survive the impact.
You need a computer capable of calculating the impact forces within margins, a computer you could conceivably later use as ablation or reaction mass, and then a number of small events.
This is because in space, things like the hydrophobic ejection of spore capsules happens at insane velocities and does not get the ejection "broken" by the thickness of the atmosphere.
Imagine a spore container several "spores" thick. The outer container uses hydrophobics to eject an inner container that contains some water and spore containers, and suddenly you have the means to get going well beyond the speed of sound with an utterly tiny package.
Remember that there's a cube/square problem going on, where the smaller you can make the interception device, the less energy it takes to intercept at speed.
Then, having an impactor made to take the structural collapse and damage for the internal payload would complete the journey.
I will maintain that having several hundred thousand tons of material to add to or modify and which is already going somewhere, anywhere, is a much better way of having several hundred thousand tons of material launched from a planet and accelerated yourself.
Then in a few hundred thousand years (what does it matter to the probe systems? They hibernate!) when something happens you wake up and check if you have found yourself somewhere interesting, maybe shed some probes, and continue on through the universe.
In fact this particular interstellar object has its own 'coma' or micro-atmosphere as well as frozen gassy deposits which would be VERY nice to have for future probe applications, since not only are they easily accessed reaction mass, they are also useful hydrocarbon sources for whatever other features might need to be farmed up.