I'm far and away from being a Scott Manley, but it seems if you entered into a retrograde orbit around Eve, with one end of the orbit matching up with the innermost section of Glily's SOI, and the other end of the orbit matching up with the outermost section of Gilly's SOI, the small but opposite course corrections you are getting would cancel and you could have an infinitely stable orbit that changes SOI twice per orbit (well 4 times, into and out of Gilly's SOI twice). Sounds like an existential hell setting it up though. Might give it a shot when I get home from work lol.