Valkyrie Helmet Guide

Enter a game of Cursed mode in Tier 1+ 

The cat cafe side quest requires you to pick up 5 toy mice around the map, and also scare 5 cats around the map.
To pick up the mice just interact with them. To scare off the cats when you find one just stare at it for about 5 seconds. Do take note that all mice will be spawned in at the start of your game but there will only be 1 cat, when you scare off the cat after 2 rounds have passed a new cat will spawn in for you to find.
You can pick up all 5 mice at any point you like but only scare 4 cats away until a later step.

For a detailed guide on all the locations of the mice and cats along with images please reference this gamerant guide:
Cat Cafe Quest

IMPORTANT NOTE: Later on to complete the relic requirements you MUST get an Assault Wave at the Central Courtyard location. The only way to guarantee this is to fail any of the Assault Waves that are not at that location throughout your game. This is optional to do but is highly recommended. If you fail an assault wave fully the area will permanently have all fissures pouring out smoke and the banners must be reclaimed to stop it but DO NOT reclaim the banners otherwise the future Assault Waves can be at those locations again.

Here is a video guide if you prefer to see and hear exactly how all this is done: MrRoflWaffles Guide
(If you know how to do these skip this step)

CRAFTING THE MANEKI-NEKO
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Next you must craft the Maneki-Neko. To do this you will have to gather 3 parts in any order.

First is the Furin (look like a hanging bell):

Second is the Cat Figure:

Third is the Karakuri Figure:

Assemble the Maneki-Neko at the worktable that is just to the left of Vulture Aid in the Workshop.

OBTAINING THE NEKOMANCER WONDER WEAPON
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Now that you have the Maneki-Neko you must get the Nekomancer wonder weapon.

Here is a list of things to do to acquire this:

To update the Maneki-Neko you must grab 3 parts that are always in the same locations:

A piece of coal in the oven in Kitchens
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A matchbook on a shelf by Double Tap in Kitchens
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White powder behind the Riflemen trap at the central courtyard. To get this one you have to go stand behind the trap and just to the right of the doorway that leads you behind the trap, reload your gun and the powder will appear.
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Once you have all 3 parts, hold interact at the table where you originally crafted the Maneki-Neko and you will receive the upgraded version.

NOTE: It is speculated currently that the white powder may not be behaving correctly and that you are supposed to upgrade the riflemen trap to get it the correct way but this is not confirmed. It may however be possible that a patch rolls out in the future that makes it so you cannot get the white powder as stated in this guide.
Credit to Doughnuts Youtube video guide for the screenshots.

You must now get 4 plate parts that have to be repaired.

To acquire the parts use the upgraded Maneki-Neko equipment and while you are standing in its safety circle kill zombies around you with the charged shot of the Nekomancer wonder weapon. Each time you do this there will be a chance that where you had the Maneki-Neko on the ground there will now be a plate piece. I believe death perception can highlight these pieces to make them more visible.

Once you have all 4 plate pieces, place them down in the Kitchens in the firepit that is to the right of the Mystery Box location. Then throw a molotov on top of the plate shards and once the fire is gone the plate will be repaired and can be picked up.

NOTE: Getting the 4 plate pieces seems to be random and can require you to use a lot of the Maneki-Neko equipment. If you have "Picky Eater" for Vulture-Aid it will help you get more equipment, also if you have "Fully Equipped" for Mule Kick that will help too.

Now that you have the plate you can finish the cat cafe side quest from earlier. Simply locate the last cat and scare it away.

Now if you go into the Tea Garden and look on top of the roof of the building that is across from the mystery box you will see the 5 cats sitting up there. Just stare at them for a few seconds and your screen will flash white and a bunch of things will now appear in the building.

There will be a cat tree that is now in the building and on the cat tree will be a pile of cat nip you must pick up. Once you have picked up the cat nip, go to the Central Courtyard and there is a rock that is in front of the Mystery Box location. Climb on top of that rock and look at the center of it and interact with it, this will place down the plate with catnip onto the rock.

Play your game now as normal until you get an Assault Wave in the Central Courtyard. As recommended earlier if you failed the other Assault Wave locations you can guarantee this to happen.

Once the Assault Wave is going you must get kills using your Nekomancer Wonder Weapon. As you kill zombies you should see red souls leaving them and going to the catnip plate. Once you've gotten enough kills you will hear Mr. Peeks laugh and the relic trial portal will open.

NOTE: It takes quite a lot of kills to get the portal to open, if you need extra zombies to kill let the zombies destroy the statues part of the assault wave and when they move onto the banners you will get extra time and more zombies. You can use decoy grenades to easily keep these extra zombies in the Central Courtyard for killing.

The relic trial portal can be found in the Training Area on the wall next to the Ammo Cache.
The relic trial challenge is that every time you reload you explode and take some damage.