I don't think a left handed bat is that necessary.
For whoever we were to acquire, the barrier to crack the starting lineup is someone better than Gurriel/Hernandez in the OF or Kirk at DH. That's a damn good bat and would cost a lot. But if we aim lower and go for a bench bat, said bat needs to be better than Raimel Tapia or Cavan Biggio, both of whom are hitting better than league average.
Sure, if a superstar like Soto is involved I'm all ears but otherwise we need to use our bullets on pitching.