Would I be able to buy that small plastic part only?

[magictrampnj]

(no luck in HD/Lowe's nor nearby small shop so far)

 

[MicEd69]

Probably not. But it looks brand new. If so, call the manufacturer and they may be able to send you either that piece or a new kit. I had a Peerless faucet that broke when I dropped a tile on it when I was tiling the shower wall. They sent me a replacement kit at no charge.

 

[Twowaxhack]

That’s Kohler, it usually comes with their $12 stem I think. Or you can call kohler.

Here ya go

KOHLER GENUINE PART GP1074231 SPLINE ADAPTERS , Black
https://a.co/d/4RZlNNU

 

[Twowaxhack]

You might can find this kohler stem at Home Depot. It usually comes with the spline adapter.

The spline adapter should fit your application as well.

Kohler K-GP77005-RP Ceramic Valve, One Size, Rough Plate
https://a.co/d/dduWQIu