"So you need about 100 meters thick of ice to function as heat shield and source rocket propellent.
The area could be about 4 sq km, so this mass would be about 400 million tonnes. Add about 10% of this mass for tankage, plumbing, pumps, rockets, and 20% as payload,
so less than 450 million tonnes in total mass."
Oops.
400 million tonnes of ice
40 million tonnes of rocket "dry weight"
80 million in payload
Total: 520 million tonnes

The 80 million tonne payload could include a different
type rocket or propulsion system- e.g using solar sail.
And it's also possible to "recover the first stage" to use it again in the Sol system. With another propulsion system
one might add another 100 km/sec. Leaving you with total velocity of about 1/1000 the speed of light. And therefore requiring about 1000 years to leave Sol, a few thousand years to get to another star system and another 1000 years to enter that star system.

So, this is something that could done within as soon as 50 years. And increasing the escape velocity by a factor of 10, so instead of thousands of years it's hundreds of years, is probably possible. But it's likely that if the difference is hundreds of year compared to thousands of year, making a larger payload [and therefore having more usable resources to use along the trip] would more desirable than getting to star system faster.