Tiny Grains
A 5* Kinetic Shotgun.
Fenny charges at enemies a lot, and spends a lot of her time very close to enemies, meaning she can usually proc Tiny Grain' proc very easily.
Even without the buff, Tiny Grains' high stats mean it is strong, and it is Lionheart's strongest weapon.
Maritime Scout
A difficult-to-get weapon, but it's Ballistic, Standard Skill, and Ultimate Skill DMG buffs are very strong, and make it Lionheart's best 4* weapon, and second best to Tiny Grains.
If you aren't proc'ing Tiny Grains buff, then Maritime Scout performs somewhat similarly, with the only real difference between the two being base damage - Tiny Grains has higher base damage (811 vs 645, or +25% more).
Full Alert
Gives <Operative Level x1> every second, up to 10 stacks.
At Level.80 this is a +800 ATK buff, which is a good passive buff, although Full Alrt isn't quite as good as Maritine Scout.
Precious Baby
Fenny does make a shield quite frequently due to her standard skill, so Precious Baby's 24-40% extra shield construction efficiency can be a useful stat for her to improve her survivability. Extra survivability is usually not needed though, and Precious Baby is a minor buff.
Given that Maritime Scout and Full Alert can be difficult to get, Precious Baby can be the best/only 4* Kinetic shotgun you have, which makes Fenny's the best option by-default.
The game recommends Mechanized Cop for Fenny, and whilst it is a Kinetic Shotgun and thus does work well for her, its 1-shell capacity means that you have to reload after every shot, which gets real annoying real fast, especially as it doesn't have a particularly fast reload speed.
Its very high Compatibility stat means it does deal more damage per shot (about two-thirds more damage than that of a regular 4* shotgun), but its constant reloading means its dps isn't actually that great - It deals around 1% more actual dps than Full Alert, and around 2% more than Precious Baby. Factoring in it's effect, Mechanized cop on-average deals less damage than Maritime Scout unless you are killing enemies very frequently to proc Mechanized Cops effect.
Mechanized Cop is, technically speaking, a good shotgun, and it's perfectly fine to use it, but the advantage it has over other 4* shotguns is tiny, and given how annoying it is to play I don't recommend it - just use Maritime Scout or Precious Baby.
If you don't have Lionheart's first Neuronic, then Lionheart doesn't gain any benefit from a Kinetic weapon specifically, and as such if she doesn't have that Neuronic then she can use any shotgun just fine, in which case other shotguns like Neon Tube and Discordance are just fine.
I do not recommend using Fenny as a support, but if you did want to use her as one:
Lapis Lazuli
When tFenny uses her support skill, Lapis Lazuli grants +ATK to the deployed operative for 15s.
Lionheart's Support Skill has a 25s cooldown, but if you have her Neuronic then hitting enemies can reduce its cooldown, and Skill Haste on her logistics can reduce it further, which can allow Lapis Lazuli to have 100% or nearly 100% uptime.
Rock Python
The ATK buff if the target is at full HP can be a nice bonus, if you can proc it.
Whilst it's not a large bonus, combined with Fenny's M5 bonus and a support-based logistics squad then Fenny can give out several buffs to her team, which add up to a decent amount, although it's not as good as Lapis Lazuli, nor as consistent.
For weapon Parts, just go with ATK.