In Terminator 2 Judgement Day, Uncle Bob(Arnold) says it can't self-terminate. In Terminator 3: Rise of the Machines, we see a mushroom cloud when Arnie tosses its' damaged power cell and the cell explodes. T3 Arnold later contradicts the Terminator in T2 when T3 Arnie self terminates itself and the Terminatrix with T3 Arnie's second power cell, so we know Terminators can self-terminate to accomplish their mission. Was there an in-universe reason why the Terminatrix didn't destroy itself by exploding its' own power cell when Kate Brewster, the last of John Connor's lieutenants, was near by? Was there more to the Terminatrix's mission than killing John Connor's lieutenants?
I'm not sure where you have the information that Kate Brewster was "the last of John Connor's lieutenants" from?
T-X Wiki lists (with no explicit detailed references):
11 named termination targets, starting with John Connor as primary target
22 human targets (statement from T850) total aside from Connor/Brewster
4! named targets that were terminated (aside from collaterals)
Therefore, in the absence of conflicting primary information, we should assume that Kate Brewster was far from the last on the list at the moment T-X met her.
Probably even more importantly, John Connor was a primary target. There is no way T-X could have completed the mission without finding and killing him