Ever since the intro said it was pretty much impossible to Nuzlocke the elite four, I have been tinkering with ways to get through it without losing any party members and minimal hax on set mode. I think I finally found one. I also tried not to use pseudo-legendaries in this team, to make it a bit more of a challenge, but ultimately having a good steel type, is critical, and none do the job nearly as well as Metagross
The team is
Metagross - attack boosting nature/high attack iv
Slaking
Charizard
Hariyama
Ludicolo - rain dish
Milotic
Other pokemon that seemed viable/worked well: Swampert, Slowking, Arcanine, Bellossom, Claydol, Gardevoir, Glalie
Sidney: It is just a matter of timing your switches right. For me, he will always use his Machamp after his Sabeleye. If you start with Metagross, he will try to EQ it. So you switch to Charizard, then he will try to rock slide, so you switch to Hariyama, then you fake out and cross chop/superpower. A similar switch is required on Tauros, but otherwise, the battle is pretty simple.
Phoebe: This battle you can even get some hax against you and still win without losing Pokemon. I was hit by hypnosis and missed some attacks due to double team but still managed it with ease. Having a heightened attack stat on your Metagross is important because then you can 2 shot Sabeleye, which is helpful to prevent Sabeleye from setting up more double teams.
The Glacia battle becomes ridiculously easy with this team, the only problem is making sure you don't get hit by the explosions and self-destruct. Even if you do, you can still win. For example, if you switch in Metagross, when Wailword self-destructs it will survive. It never happened to me with Regice, but I am assuming it could take that too. Ludicolo is an absolute gem in this matchup. The combination of rain dish and giga drain, makes it recovery extremely strong. Using this strategy, dewgong, swampert, and lapras are KO'd incredibly easily. Then between, Hariyama and Metagross, the rest of the battle is pretty straightforward.
Lance: His Latios is really really annoying. I think I spent most of my runs trying to figure out a strategy to work around its draco meteor. It will one-shot pretty much anything you throw at it. Ultimately, Milotic is bulky enough to tank a hit and hit back with an ice beam. But, that still won't KO! Instead, use mirror coat. Another strategy is to just soft-reset till you freeze with ice beam. Metagross doesn't work as a lead because it's meteor mash can't 1 HKO without a crit. Outside of his Latios, the rest of Lance's team is fairly straightforward tho. Having an ice punch on your Ludicolo is helpful.
Steven: Personally, I found Lance to be harder than Steven because of that over-powered Latios. Though, this battle is incredibly difficult without a fire-type because Jirachi and Metagross are so bulky. But Charizard can OHKO both of them with ease. From there, you just should have a good amount of priority moves on your team as Deoxys will usually weaken itself into range, for one priority move to KO it. Arcanine's extremespeed is extremely useful for that, but since it provides less value in switching in on earthquakes, I still think Charizard is better.
The last thing I'll note is that Slaking is incredibly powerful. On almost all my runs, Slaking picked up the most amount of KOs. Since it has great speed, after the turn it switches in, it usually moves first. Moreover, it is so powerful, that it can often OHKO a lot of foes. Even when it doesn't, just by smarty cycling switches, pretty much no pokemon can take two attacks by Slaking.
I understand that a lot of these pokemon are rare encounters, so it is hard to get them on your first encounter. But the Elite Four's teams are all so punishing and broken, that you do need the perfect team, in my experience. Put together, this was a really fun exercise, and I hope this helps anyone trying to beat the Elite Four without losing any party members. If any of you have tried something similar or have built a team from more common encounters that can beat the Elite Four without losing a member or lots of haxs, I'd be really interested to see it!