You want weapons that have Maliwan (Element Switch Underbarrel). Ideally, have these be many flavors of the 
Mantra.
Licensed Parts
- Jakobs (Ricochet Accessory) is very strong: it enables ricochets and access to powerful enhancement effects.
- Ripper (Charge Mag) is an extreme DPS boost, but is generally reserved for bossing due to the charge time and ammo inefficiency. The more fire rate PARTS a Mantra has, the quicker it will charge-up the Ripper magazine.
- Mantra can roll both of these licenses at once.
Avoid most bladed weapons, except maybe for Bladed 
Maliwan Shotgun/Mantra. Those currently apply an extra 82.5% Melee Damage boost to your Melee Attack. However, they do have long recovery frames, which still makes them pretty meh to use.
Equip the weapons in the order most comfortable to you, and organize their modes in a way that has all your elemental matchups covered for 
Dead Ringer summoning, 
Phase Covenant attuned melee attacks and 
Groundbreaker.
Other Weapons
If your 'melee philosophy' is open to experimentation, you can also use heavy-hitting underbarrels to build up Groundbreaker, like
War Paint, 
Order Repkit, 
Triple Bypass are all good choices of Repkit with their own relative strengths.
Look for these effects, or combinations of them:
- Enrage (Experimental)
- Enrage is highly recommended if you're looking to maximize damage, since it will apply to all damage and be double and triple dipped by
Reverburnations into 
Bloodletter, and can go even further if you decide to use Groundbreaker.
- Hard Hitter (Steroidal)
- Elemental Affinity (Reactive)
- Accelarator (Adrenalized)
- Amp (Amplified) on any of these Repkits can be situationally useful if your self-imposed melee rules allow usage of heavy-hitting underbarrels for Groundbreaker.
Slippy
The Slippy gives us another way of accessing melee, and is excellent at building up Groundbreaker. It is also a safe way to burst Vile crystals enemies and some other types that you find to be very dangerous when at close distance. It will be attuned to Vex's current held element due to Phase Covenant.
It gets its own triggers of Reverburnations.
Watts 4 Dinner is actually the better shield for Slippy and building up Groundbreaker, since the 
Heavyweight 100% melee damage doesn't apply to it (it consumes the armor but does nothing). - The bonus shock from Watts 4 Dinner gets its own Reverburnations, as does
Infusion.
Look for a variant that has any of
- Crit Chance
- Nuke
- +1 Max Charge
Then pick your preferred one.
The 'Crit Knife' 
Jakobs Grenade is great if you're choosing to lean more on the Mantra for your damage, since it helps it access crit damage alongside the boost from 
Coven + Specters during bossing.
Class Mod
For stat rolls on any class mod, all of these are acceptable:
- Melee Damage; Skill Damage
- Melee Crit Chance; Skill Crit Chance
This is excellent for bossing, and also works great for general use, with a Watts 4 Dinner shield being recommended for it.
- +5 Reverburnations; +2
Fell Inscriptions or preference - This is probably the best damage you can get, period. It's a nice swap-in for bossing if you prefer other class mods for mobbing.
Fun option, won't usually have the bossing peaks of Firedancer but can be reliable.
Stick to your slams for Groundbreaker when using this class mod if it has 
Haruspex, for better consistency, since your melee attacks might be interrupted by Haruspex unexpectedly.
- +5
Grave Sights; +2 
Reawakening or preference - +5 Haruspex; +2 Reawakening or preference
Teen Witch
This is another safe general use option. Can be the main class mod if you want to play Mantra + Amp Watts 4 Dinner. It sustains both damage and survival excellently through its very powerful legendary effect.
- For a pure damage-oriented roll, +3 or +2
Corporeal and +4 or +5 
Blood Is Magic is ideal. - For best minion sustain, +2 or +3
Grave Thirst and +5 or +4 Blood Is Magic.
Watts 4 Dinner
Siphon variant is the main shield choice for the Fire Dancer or 
Misericorde class mods, be it for bossing or mobbing.
Amp variant is the main shield choice if using Teen Witch + Mantra. It's also great for stacking up Groundbreaker with strong underbarrels on any of the class mods if you're into that.
- This shield's bonus shock damage inherits the source type, and will apply to our melee damage sources. This means that on melee damage instances, including Slippy and Mantra, the Reverburnations skill can be triggered on your base damage, on Infusion and on the bonus shock from Watts 4 Dinner.
- The minion drain from Dead Ringer can trigger the Siphon part of shields for you.
Heavyweight
Decent secondary choice for true melee use with Teen Witch, since it mitigates the armor loss and allows melee attack spam at will.
Look for variants that have the Missile Swarm effect on it, ideally the double variant that shoots out five missiles. Missile Swarm is ideal since it contributes to Groundbreaker and 
Recurrence.
- Careful with Missile Swarm if you don't intend to use Teen Witch, since it can down you. Use other rolls in that case.
- You could look for different elemental variants or just settle for a neutral element like Kinetic.
- The 100% melee boost from this shield does not apply to , Mantra, Dead Ringer Reapers or Trouble.
Enhancement 
Jakobs EnhancementRipper EnhancementMaliwan Enhancement
For stat rolls on enhancement, look for:
- Shotgun Damage; Gun Damage
- Weapon Equip Speed; Shotgun Equip Speed
- Shotgun Crit Damage; Gun Crit Damage
- Shotgun Splash Damage; Gun Splash Damage
- Shotgun Mag Size; Gun Mag Size
- The 'Gun Damage' and 'Shotgun Damage' boosts are very powerful for the Mantra.
- The 'Shotgun Equip Speed' and 'Gun Equip Speed' boosts will stack, affect your melee recovery rate and also the speed of Maliwan mode switches.
- The 'Shotgun Splash Damage' and 'Gun Splash Damage' boosts work on Melee attacks and Groundbreaker, as long as you hold a weapon that fits the requirements. AVOID using both boosts together, they can cause weirdness and cancel each other out. While they're a relatively weak boost, I'm listing them here anyway since they're one of the few enhancement stat boosts for melee damage.
Quintain; Bounce Pass; Piercer
- Very good choices with high general applicability.
Multistrike and Sequencer are also good, just beware that Specters can mess up the Sequencer effect.
Entropic; 
Devourer; Hard Charger; Free Loader; Short Circuit
- Some of these can be extremely powerful, however the RNG and overall inconsistency can be an annoyance. Experiment to your liking.
Lame; Mixologist
- The weaker options of the bunch, but still serviceable.
Vex
Dead Ringer
