Sure, it's likely that we're missing details, and I agree that the LW is not a reliable narrator (and should have rehomed the dog as soon as it was clear that dog and Liza weren't compatible).

With the information that we have in the letter -- Liza doesn't want the dog around, and has been physically aggressive towards it -- and no word from LW on whether Liza thinks the new baby is great or is displacing her in LW's life? Yeah, I'm erring towards caution.