I’d go with either the Birds of Prey or the JSA. I lean towards the JSA simply because with the BoP, I would never stand a chance of being “the hot one” or “the smart one”. Plus, I’d happily accept 24-hour access to Ted Grant.
You know, I love Babs and Dinah–I really do; and Helena is all right in her own sociopathic way. But I’m not sure what they could do against Darkseid (a) scream for Power Girl or (b) bleed.
1997 and 1998. They hit the scene about the same time as Electric Superman and the series lasted 17 issues, plus a 1,000,000 issue.
They fall foul of the ‘Darkseid’s after your ass’ test, though.
If you’re LUCKY, Off-Ramp could keep the team moving a step ahead of DS. I severely doubt he has access to any realms a Boom Tube couldn’t access - at least not while the Young Heroes are still together. In YHIL 1,000,000, it’s demonstrated he’s got the ability to go through time as well, but he looked a good 30, 40 years older at that point, never displayed the ability in the 20th century issues, and apparently hadn’t seen Frostbite for years before that time, and…well, if he hadn’t seen Frostbite, we can be pretty sure he hadn’t seen the rest of the team for a while. So, there’s no reason to believe he could go into the far past or far future to avoid Darkseid, while the Young Heroes were still together.
And if Off-Ramp couldn’t keep you ahead of Darkseid?
Well…Monster-Girl’s uncle and Totenjaeger are the two most powerful enemies they fought, neither of them is on Darkseid’s level - nor were either in command of an army of Paradaemons and Kalibak - and both of them gave the Young Heroes fits.
That shouldn’t be “or”. It should be “and”, even with PG present.
The Avengers, for reasons already stated.
But I suppose I’d fit in best with the Doom Patrol. 
Hmm, if I’ve got to keep Darkseid off my back for a year then I want to join the New Gods.
Plus, hey, I get to be a God. 
And Major Bummer. And Off-Ramp and his, ah, family might just be able to pull it off, is what I was thinking. And if not, well, there’s always a team-up.
You know what? Good call. I choose the Endless instead because a) they’re practically unbeatable and b) Thanos would just end up flirting with/chasing/trying to win the heart of Death anyways. Mission accomplished!
I’m not saying PG could actually defeat Darkseid. But she could keep him busy long enough for Babs and Dinah to … um…
Damn.
The Defenders.
Dr. Strange, Hulk, Sub-Mariner, Silver Surfer, Valkyrie, Nighthawk–a fun group.
X-Men: Used to be that you’d just be feared and hated. Now you’ll be brainwashed by HYDRA, cloned by Mister Sinister, bodyswapped by the Hand, possessed by cosmic entities, die, come back, die again, have your memories erased by the Weapon X program, Psylocke will steal your man, and you’ll discover the angsty, teenaged, genetically-engineered alternate timeline offspring you never knew you had. Count me out.
Avengers: As Deadpool says in Cable & Deadpool #24, “They let anyone be an Avenger now! I heard Black Talon is an Avenger!”
Fantastic Four: I hear the pension sucks.
Champions: Remember in Marvel Team-up when Nova gets all excited and says, “Hey guys, we work together so well, we should become the New Champions!” and Spider-Man, Moon Knight, et al, are just like, “Eh, no.” That’s what being a Champion is like. A whole lotta eh.
Ben Grimm just cashiered out recently, to the tune of over a billion dollars. I’ll take that over my actual plan.
Wait…maybe we’re going about this the wrong way. The OP said we’ve pissed off the “local equivilant of” Darkseid or Thanos or whoever…so, by my thinking, if you throw in with some powerful, no-nonsense kickass team—like the Authority, or the Ultimates—your cosmic enemy is going to be an appropriately utter badass.
However…if you choose a lame team from a lame universe, the local “Darkseid” probably isn’t going to be some 'orrible, omnipotent archvillain. He’s going to be lame, too. Maybe less so than the heroes or the rest of the villains, but he’s still going to be a long way from a Great Old One who gets off (literally) on murdering anime horror villains.
So, under that rationale, I might throw in with the Superfriends. Or Mighty Mouse. I might not be able to look myself in the mirror after a few weeks, but I’d probably live longer.