I am going to go a little bit against the grain and say definitely check with the therapist first, because given her history it's possible that "abandoning the dog at a shelter" would set off all her insecurities about whether you would do that to her if the baby wanted her gone - some of this may be an ongoing test to see how conditional her welcome in your home really is.

But six years is a very long time for you to have forced Liza to live with a dog she didn't want and doesn't get on with that barks and pees everywhere and is scared of her, so if you consult with Liza and the therapist, and if they both agree this is a good idea, then you 100% pick Liza over the dog! (Also tbh it doesn't sound like you two really like the dog that much either?) You should involve her in every step of finding the dog a new home in the way you didn't involve her in getting it and keeping it, though, so it doesn't feel like the two of you are kicking the dog out but like the family is doing something they agreed on together.