T1 Hulls
To be honest, the Amarr ships aren't bad on paper. There's nothing you can't theoretically use to dive into the Abyss up to tier 2, and maybe tier 3.
Arbitrator
This looks like a decent ship for clearing tier 1 and tier 2 dungeons. Tier 3 you'll lose it quite easily to RNJesus.
[Arbitrator, Diver]
Medium Ancillary Armor Repairer, Nanite Repair Paste
Medium Armor Repairer II
Damage Control II
Energized Adaptive Nano Membrane II
Drone Damage Amplifier II
10MN Y-S8 Compact Afterburner
Medium Electrochemical Capacitor Booster I, Navy Cap Booster 100
Stasis Webifier II
Stasis Webifier II
Rapid Light Missile Launcher II, Nova Fury Light Missile
Rapid Light Missile Launcher II, Nova Fury Light Missile
220mm Vulcan AutoCannon II, Fusion M
220mm Vulcan AutoCannon II, Fusion M
Medium Anti-Explosive Pump I
Medium Capacitor Control Circuit I
Medium Capacitor Control Circuit I
Hammerhead II x5
127 EHP/s active tank cap stable with an AAR burst tank up to 406HP/s on overheat. Can you spare a minute to reload the AAR? Not in some situations, so I expect you'll lose a few of these. DPS is a flat 400 as fitted, but mostly drones. RLML's may not be the best for sustained DPS but you will definitely rip through the swarms of drones. This is the pick of the litter tbh.
Maller
If you want to do tier 1's only...
[Maller, Diver]
Damage Control II
Energized Adaptive Nano Membrane II
Medium Armor Repairer II
Tracking Enhancer II
Tracking Enhancer II
Heat Sink II
10MN Monopropellant Enduring Afterburner
Medium Capacitor Booster II, Navy Cap Booster 200
Stasis Webifier II
Focused Medium Pulse Laser II, Scorch M
Focused Medium Pulse Laser II, Scorch M
Focused Medium Pulse Laser II, Scorch M
Focused Medium Pulse Laser II, Scorch M
Focused Medium Pulse Laser II, Scorch M
Medium Auxiliary Nano Pump I
Medium Energy Burst Aerator I
Medium Capacitor Control Circuit I
Hobgoblin II x3
Fatty McBaitship does 300 DPS, which is not enough to really rip through the drone cruisers and battleships in tier 2. The tank is also shite, at 163EHP/s cold and you need a cap booster to achieve 8m of cap life with 200's, and you can be stable with 400's. Luckily you have lots of cargo for those boosters and the loot you won't be getting.
Omen
This is a fit very similar to The Gentleman Gamer's. He did a tier 3 dungeon in one, but he didn't get 5 Striking Leshaks, which would have rinsed him instantly. So many people have said the test server dungeons were easier, so who knows?
[Omen, Diver]
Medium Armor Repairer II
Damage Control II
Capacitor Power Relay II
Energized Adaptive Nano Membrane II
Heat Sink II
Reactive Armor Hardener
50MN Cold-Gas Enduring Microwarpdrive
Stasis Webifier II
Large Compact Pb-Acid Cap Battery
Focused Medium Pulse Laser II, Scorch M
Focused Medium Pulse Laser II, Scorch M
Focused Medium Pulse Laser II, Scorch M
Focused Medium Pulse Laser II, Scorch M
Focused Medium Pulse Laser II, Scorch M
Medium Capacitor Control Circuit I
Medium Auxiliary Nano Pump I
Medium Auxiliary Nano Pump I
Hammerhead II x2
Hobgoblin II x3
357 DPS, which is enough for tier 2 but will struggle with tier 3 if you get battleship spawns. But that won't be your problem, it'll be the 146 DPS tank. Plus you'll be stuck running electrical or firestorm effects thanks to your lasers. You will also have to kite. A lot. And probably pop drugs. Still, it's cheap.
Navy Cruiser Options
Navy Augoror
[Augoror Navy Issue, Diver]
Damage Control II
Medium Armor Repairer II
Energized Adaptive Nano Membrane II
Heat Sink II
Heat Sink II
Tracking Enhancer II
Tracking Enhancer II
50MN Cold-Gas Enduring Microwarpdrive
Medium Capacitor Booster II, Navy Cap Booster 400
X5 Enduring Stasis Webifier
Heavy Pulse Laser II, Scorch M
Heavy Pulse Laser II, Scorch M
Heavy Pulse Laser II, Scorch M
Small 'Vehemence' Shockwave Charge
Small 'Vehemence' Shockwave Charge
Medium Nanobot Accelerator I
Medium Nanobot Accelerator I
Medium Capacitor Control Circuit I
Hobgoblin II x3
See Omen, above, above for commentary. 388 DPS, 131 DPS tank. Smartbombs can be good for clearing the Striking Vila Damavik drones, or just popping the can as you orbit at 2,000m.
Omen Navy Issue
[Omen Navy Issue, Diver]
Damage Control II
Energized Adaptive Nano Membrane II
Medium Armor Repairer II
Capacitor Power Relay II
Heat Sink II
Heat Sink II
Heat Sink II
Medium Electrochemical Capacitor Booster I, Navy Cap Booster 400
Stasis Webifier II
50MN Cold-Gas Enduring Microwarpdrive
Heavy Pulse Laser II
Heavy Pulse Laser II
Heavy Pulse Laser II
Heavy Pulse Laser II
[Empty High slot]
Medium Auxiliary Nano Pump I
Medium Capacitor Control Circuit I
Medium Capacitor Control Circuit I
Hobgoblin II x5
Warrior II x5
You put chrome rims on the Omen, you get this. 428 DPS, 109 DPS active tank, run Forrest, run. Maybe tier 2 material.
Tech 2 Cruiser Options
Generally, I'd say no for a Zealot. It's basically a Navy Omen, so use the Navy Omen. Curse and Pilgrim, no. So you're left with...
[Sacrilege, Diver]
Medium Armor Repairer II
Medium Armor Repairer II
Ballistic Control System II
Imperial Navy Energized Adaptive Nano Membrane
Ballistic Control System II
Large Compact Pb-Acid Cap Battery
10MN Afterburner II
Stasis Webifier II
Stasis Webifier II
Heavy Missile Launcher II, Inferno Fury Heavy Missile
Heavy Missile Launcher II, Inferno Fury Heavy Missile
Heavy Missile Launcher II, Inferno Fury Heavy Missile
Heavy Missile Launcher II, Inferno Fury Heavy Missile
Heavy Missile Launcher II, Inferno Fury Heavy Missile
Medium Proton Smartbomb II
Medium Auxiliary Nano Pump II
Medium Auxiliary Nano Pump II
Hammerhead II x5
This did me quite OK, until it didn't. I'll blame Delonewolf because I can't blame the fit; it tanked fine, just not 5 battleships fine. Would throwing more pimped reps on it have helped? No, because the chances of bashing down 5 Striking Leshak BS and finishing two other rooms in 20 minutes is dicey.
This really brings home the fact that billion ISK fits aren't the be all and end all, just like I said pre-launch. The Abyss will just throw a bad roll of the RNJesus dice, and you'll die. So are you going to toss a billion ISK at it? Or 300M ISK? That's a discussion for later.
Finally the Devoter....no. No advantages over the Sacrilege bar the 18km scram (if that's useful) with less DPS, less tank, and costs more.
So that's it. If you want to be restricted to running 2 of the 5 environments and want to basically get railroaded into the Sacrilege and lose one a week at least, then Amarr is your race!