Live 1-3

Hero opens HJ to $11 w 5s5h 150bb
Bttn calls and is a splashy spot with a very wide range 40bb
BB calls and is a solid winning reg 150bb

Flop Kc23c
Pot = $30
BB checks
Hero cbets $15
Bttn folds
BB calls

Turn Ko
Check check

River 4o
BB bets $40

In hand I didn't consider it, but looking back, this hand seems just about the best bluff candidate I can have, blocking 56 and A5, and I also do not have the 5c in hand, which makes for more bluffs in his range, if only slightly(maybe a hand like Q5s that gets defended pre and bet folded)

Anyways, Ive given this some thought, and think I should either fold or raise here, but I don't really have a good idea on what sizing is appropriate, and why it makes sense from a theoretical perspective. Can anyone help me make sense of this? I want to just say between 4-6x, because it's big, but I don't have a solid reason for WHY.

spoiler too, I actually called in the spot, which in hindsight is probably a bit too loose