What's the benefit of keeping a Shadow Pokemon?

There isn't one, unless it fits one of the typical min-max desired attributes:

• Has high IVs
• Has desired moves

Other than that, there's no point in keeping them as they are also fodder Pokemon, or if you intend to at some point mass-purify them.

Just a note: Fodder shadow is more valuable than regular fodder Pokemon as they have lower costs when upgraded using candy (like Lucky Pokemon) once you purify them, so if moves and IVs aren't your goal: That's a reason to keep them (ie. a shadow Bulbasaur with bad IVs versus a regular Bulbasaur with also bad IVs)

UPDATED for Shadow Pokemon gaining +20% Atk and -20% Def

The benefits from capturing Shadow Pokemon:

• Shadow Pokemon have their Attack increased by 20% and their Defence decreased by 20%. This is a huge buff for raids - clearing the boss as quickly as possible is more desirable than survivability, both increasing the chance of winning before the timer runs out and increasing the chance that you get better rewards. This makes Shadow Pokemon that are high performers in raids very desirable. Sometimes even a 0/0/0 Shadow Pokemon will deal more damage to a raid boss than a 15/15/15 Purified Pokemon! For PvP, this may also be desirable depending on breakpoints and bulkpoints, which is Pokemon specific. Commonly, but not always, Pokemon with high Attack stats generally benefit from being Shadow such as Victreebel and Gardevoir. The changes to raids and PvP are described in more detail on Game Press.
• Rewards obtained from beating Team Go Rocket such as Stardust, items against leaders and Giovanni, and clearing research tasks.
• Badges as per the OP
• Collection. In a similar vein as to how people enjoy having different forms of other Pokemon, including all the Pikachus, one can have a collection of all the different Shadow Pokemon.
• The option to show off your Shadow Pokemon in gyms and as a buddy.

Shadow Pokemon can be purified, which gives you the potential benefits of:

• Less stardust and candy to level up. This can potentially more than offset the cost to purify the Pokemon.
• The move Return, only available upon purifying. It is a good normal-type move offensively in PvE, being the charge move with the highest damage based on time and energy for Snorlax in PvE. In PvP, the move is usually undesirable due to its high energy cost, but it is an ideal move to use with some Pokemon like Wobbuffet (it and Mirror Coat are the only charge moves it can have) and Sableye (Foul Play is good, but its other charge moves are bad).
• A 2/2/2 IV boost
• Collection, similar to Shadow Pokemon. Each Pokemon also has a Pokedex category for the number of Purified Pokemon.
• To show off Purified Pokemon after depositing it in a gym or setting it as a buddy.

I'm ignoring Frustration as a reason since that move is abysmal both in PvE and PvP.