For an interstellar probe I think a space station at L-1 would better in all cases than a space station in LEO.

And the article didn't seem to call for a big heavy lift. Though I wasn't sure what an "ALS class rocket" was [some Altas variant??].
Generally speaking I think that rocket able to lift say 75 tons to LEO is about largest rocket "needed". And that the present lift capacity of Altas V and Delta-IV heavy [or Proton and Ariane V] are adequate to open the space frontier.
If it was up to me, we would go to the Moon [and possibly Mars] using EELV.

I think if you want a interstellar probe in the relatively near term, one should look at how to get the most velocity from gravity assists. And for an interstellar probe to Alpha Centauri, according to reference you linked, it needs a 61 inclination trajectory relative to solar plane. Which seem to indicate to me that Jupiter and/or Saturn will be involved in the path of the trajectory, somehow.

The approach I would suggest in regard to a interstellar probe is focus on near term use of the probe and in sense discard it by sending to to Alpha Centauri. In other words have radio telescope platform which could be used at the Earth/Moon system and then move it to inner system and then outer planets and onward to outer part of our solar system.
A sort of repeat of Pioneer probe, but much bigger and faster. And sort of like Hubble in sense of being designed to serviced or upgraded after it's in Space.
So, it's assembled in space at L-1. It's used at L-1. It's modified/upgraded at L-1 [or used and modified/upgraded in L-2 or anywhere in high earth orbit]. Then it's sent to inner part of solar system [and gaining velocity using gravity assists]. It's used while in inner solar system and it still can be modified and upgraded. It then goes to Jupiter [and/or Saturn] and changes inclination. And then it's sent back to inner system for further gravity assists and the final leg to Alpha Centauri- with again a possiblity of upgrading and improvement, before hitting it's fastest velocity.
Here you have system which will be thoroughly tested and can have lesson learned which can be added in order to improve it.

And "works" in regards of political support because near term use and NASA bureaucracy has long term cash cow- it gets yearly funding regarding various types of science mission which can used related to this mission [tacked on].
So the "thing" that finally leaves our system to get to Alpha Centauri could be a monstrosity of bureaucratic achievement.