"Honestly, I think the dialogue is suggesting that the trauma which blew the other crystals put a crack in the 4th one (suggesting 4 crystal assemblies). But your explanation is also quite valid - maybe that's why it is the spare crystal?!"
I think it literally is a spare. According to "Elaan of Troyius" they needed at least one dilithium circuit (substitute) to have warp drive capability. Looks like that's the minimum to be capable to get to the nearest service station for a replenishment.
"The Alternative Factor features 4 Dilithium "paddles" which (to my mind) echo the 4 crystals in Mudd's Women.
Finally, Kirk negotiates for 6 crystals with the miners. Either the Enterprise has the capacity to run on 6 crystals, or he was thinking ahead to the next incident, and wanted spares!"
Makes perfect sense to me. The Re-Amplification Chamber seems to have space for 12 "paddles" (the 6 drawers seem to match the 6 rows of indicators on the control panel's right side) of which four seem to be able to be re-amplified (4 small and 4 big control knobs) or tested ("experimentation chamber"?).
Since the episode suggests these "paddles" are in widespread use in the known galaxy (they are perfectly compatible with Lazarus' craft) the Enterprise
may also have the capacity to re-amplify dilithium circuits of other ships that require assistance but don't have onboard re-amplifying facilities.
Of course, we can't exclude the possibility that Kirk wanted 6 crystals because he knew that the cutting process to bring these crystals into the universal paddle shape usually results in the loss of 1 out of 2 crystals.