Shadowheart is extremely forgiving for a Sharite (which is not unique, Shar is good at convincing people that should know better that she's got their best interests in mind.... re: Sharess having a dalliance with her and nearly getting consumed as a result) but she IS a Sharite who was raised in a rather unforgiving and harsh environment. Her background and upbringing is at least as harsh as Lae'zel's just with different priorities (Sharite teachings lean more toward subtlety, Githyanki lean more toward overt domination) and she views the Gith as enemies.
It makes sense for her to encourage shooting the dog before she has a chance to become friends with her to spare the heartbreak of future betrayal.
