There are several possible ways to go, but given current science and technology, a generation ship capable of interstellar travel is not an option.
Using the rest of the question, and stipulating the billionaires are interested in taking 500 or so of their closest friends to Neptune, we can look at conventional launch and "unconventional" launches.
For a conventional launch, we will need to lift and assemble the ship in as few launches as possible. Elon Musk's project using a massive booster to loft a 500 ton interplanetary ship is the model to use, but since we are limited in time and resources, a few modifications are in order:
Revive the Sea Dragon "Big Dumb Booster". It is far cheaper (built to shipyard tolerances, welded out of sheet steel and using "blow down" pressure fed engines), and only needs a large lagoon to launch from. This makes the launch site much less vulnerable from attack as well.
Musk's interplanetary craft is powered by chemical rockets. This is ridiculously inefficient, and should be replaced by one of the many nuclear or electric drive proposals already in advanced development
Each passenger or cargo ship is going to need a lot of reaction mass to get "out there" in any reasonable time (especially if they are hauling tons of building supplies and life supper consumables). Consumables and supplies can also be sent on large pods as well. Each ship will meet up with one or more external fuel tanks (in the manner of the space shuttle) boosted into orbit on a Sea Dragon. This means the spacecraft will have two or more 550 ton tanks with extra reaction mass or other supplies attached.
Interplanetary craft in flight. For this sort of journey, picture several large "external tanks" strapped on to carry extra supplies
Of course, there may be a need to have a more capable fleet of ships, in which case, the building consortium can try to revive the ORION nuclear pulse drive concept. This has the advantage of requiring fewer lifts, and the launch site is literally defended by nuclear explosives (the ORION pulse units). ORION craft as large as 4000 tons were studied in great detail (the 4000 ton ORION was designed as a space battleship, but the internal volume can be used for more constructive purposes).
The advantages here are:
The ORION ship is conceptually much higher performance than almost anything else known to science. Launch can take place at any time without waiting for synodic launch windows, for example.
Since it is built and equipped on the ground, the quality control is better. The ship itself will be much more robust than one launched from a chemical rocket, and have much better shielding and even room for crew amenities (the original ORION team boasted they could bring barber chairs on board if they wanted to).
No need for extra fuel and supplies to be boosted into orbit to meet the spacecraft prior to setting off.
Anyone silly enough to try to riot at the launch site will be vaporized on liftoff. The fact that the nuclear pulse units are small atomic weapons means there will be a very high level of security at the manufacturing plat and launch site.
Longer term, the experience of building and operating ORION ships will translate to building bigger and better ships, eventually culminating in "Super ORION" craft capable of interstellar flight.
The downside is getting the permissions to build nuclear pulse units in the first place, and launching from Earth (unless the real reason they are leaving is existential, such as the imminent destruction of Earth).
ORION battleship in flight. A transport ship would have the internal volume used for passengers and equipment
So from a practical POV, setting up an assembly line near the shore, building an artificial lagoon for towing out the Sea Dragon and launching it, and using a series of Sea Dragon launches to lift the passenger ships and the supplies is probably the best way to go. Using a bit of a handwave, I suspect that in terms of time and resources, 3 actual spacecraft each attached to 3 pods carrying 550 toms of fuel and supplies should get you everywhere you want to go in the Solar System, so a total of 12 Sea Dragons, 3 spaceships (probably the most expensive part of the project) and 9 "ET"s for fuel and supplies.