Unofficial Challenge

Another idea (not trying myself though, anyone reading this is free to try): Setting a crafts mass to just above 0 and pushing it to a super high speed using pistons. You could tinker the pistons to make them stronger, longer and faster. Launching on Luna would make it easier, less gravity, no air resistance and easier to get into a stable Droo orbit. Have fun!

Submission
The only tinkered properties are torque and rpm of the motors.
How to use:
Create a new game (Luna has to be oriented correctly) and Launch in Luna Base. Then, wait until it automatically throws the "satellite" in an orbit. While it's flying, I recommend setting timewarp to 2,500 (it takes some time to leave Luna's SOI).

Edit: It should start accelerating after 15 seconds. Also, the orbit is almost circular, 27.7 by 30.3 MM.

@VectronTechnologies ah i see... that also means low mass and increased explosion power, so...
(but again, i'm not participating, just speculating)

@Zenithspeed, you can use the tinker panel, just don't mess with any of the properties that relate to drag.

@Zenithspeed, yes. Good ideas. This will be fun.

i won't participate because i suck at both building and flying things in this game, but i am curious as to how one is supposed to do this when all but one of the major options for propulsion are forbidden, said option being propellers, which of course don't work in vacuum, and therefore can't be used to complete the last part of raising your periapsis above the atmosphere-vacuum boundary
my best guess is to use explosions of fuel tanks, but idk how forceful those are and it also means needing to carry a whole load of extra weight with you
however, you did say "any method" and didn't rule out glitches (only full-on cheating with tinker panel and in-flight cheats) so those could also be a possibility