How I would do it.
Have a decent nuclear heater onboard, somewhat like a dirigible. After max hydrostatic altitude is reached, use gas for thrust. If you could heat it up to a few thousand degrees, it could make a capable propellant.
How Randall Munroe might do it.
Randal is very technical and very creative, a great combination (https://what-if.xkcd.com/157/)
You could have two cannisters, one providing the lift (helium balloon) and another at cartoon-physics pressures to just push you to the moon. If it is compressed to being solid helium3, then it is compressed to more than 3000 kPa, thenand maybe you could do something with the jet that comes fromwhen it meltmelts/boilboils/jettingjets out of the thrust cannister. Current technology does not yet provide for a cannister like this.