Unless, you are using a Gun Zer0, grabbing MMF first is generally preferred for Melee Zer0. I think your current spec is pretty good. Use of a Leg Hunter is also pretty nice for the Cooldown Bonus and buff to Fast Hands. I recommend grabbing a Ninja Com (either +5 BS, +4RS, +4 KB OR +6 BS, +5RS/KB) as a complement as well.
As others have suggested, grab and Rapier from Pirate’s Booty. While you are doing the Pirate’s DLC also grab a Slag Pimpernel to serve as your slagging tool. You can also do the ‘Grendel’ sidequest to farm for a Blue Ninja Mod. Your Corrosive Rubi should serve you well enough for healing purposes (I used a fire one while levelling). Pre-OP levels Resurgence is also a pretty good form of health regen.
What you currently need is probably a strong weapon to perform as a second-wind tool. The fire Hyperion PC is okay but far from ideal. And don’t recommend the Jericho at all. You definitely need a weapon with high burst damage for this. In UVHM, I primarily used Rocket Launchers, Tediore PC (the reload), Hyperion Shotties or Harold for this purpose. The latter should be an easy enough farm.
Shieldwise, you are likely gonna have to strategize when to pick up your Order (preferably near the end of your playthrough). You probably want to save the Love Thumper for end game given that getting a Hide of Terra can be pretty difficult. The primary reason why Melee Zer0 is more difficult to level is because of his shield specificity. He basically only utilizes Roid Shields and you ideally want one with a decently long recharge delay for your MMF chains. When levelling my Melee Zer0, I primarily farmed the Snowman or did good ol’ vendor farming in Sanctuary(which was very boring but surprisingly effective). You probably need to change your Roid Shield every 2-3 levels. IIIRC the Thousand Cuts Mission (The Once and Future Slab) has a guaranteed Roid Shield as the quest reward.
Grenade wise, you want a longbow or lobbed tesla with a low fuse time (preferably 0.7 or below). The rarity is not important so long it has sufficient strength to break your Roid Shield. Take note that you need to constantly upgrade your Tesla throughout the playthrough as you get higher level Roid Shields as well.