Take an unkillable frame, like Revenant, inaros, wukong, nezha... I only started using revenant on the assassination mission for the last two runs (with xaku's ability) and I'm surprised I never used him before for this. He will never die if you keep mesmer skin up.
Fulmin should be ok for most mobs but need to be able to kill heavily armoured mobs. I haven't used the Fulmin in a long time, but maybe a build based on hunter munitions and slash will work. Hunter munitions drops from ghoul purge bounties in Plains of Eidolon.
Final boss will probably need something like kuva hek, or I prefer the felarx. When I started, I used mostly the zariman incarnon weapons for the assassination. Since you can't take the fulmin, will also need a decent secondary or melee to deal with most mobs. Again, incarnon secondary weapons are really good for this. I used the laetum before, but now I use the Lex Prime with incarnon.
To be honest i wouldn't bother about archon hunts yet. I think when it was released, we already had access to the original incarnon weapons, which made it a lot easier.